Isotopic tiling theory for hyperbolic surfaces

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.


Introduction
The enumerative approaches of Delaney-Dress tiling theory [14] in the twodimensional hyperbolic plane have facilitated a novel investigation of threedimensional Euclidean networks, where hyperbolic tilings of triply-periodic minimal surfaces (TPMS) are used for enumeration of crystallographic nets in R 3 [56,45,55,32,51,9]. By relating in-surface symmetries of the TPMS to ambient Euclidean symmetries [49,28], the problem of graph enumeration and characterisation in R 3 is transformed to a two-dimensional problem in equivariant tiling theory. The idea is that tilings of the hyperbolic plane can be reticulated over the surface to give a Euclidean geometry to the tile boundaries. This idea has been explored in several contexts over the past 30 years, including standard hyperbolic tilings by disk-like tiles with kaleidoscopic symmetry [35,51], infinite tiles with network-like boundaries [33,34,18,17,36], and infinite tiles with geodesic boundaries [19]. The approach is motivated by the confluence of minimal surface geometry and the structural chemistry of zeolites and metal-organic frameworks [29,30,31,11].
The enumeration of such hyperbolic tilings reduces down to a problem of enumerating all embeddings of graphs on the orbifold associated to the symmetry group of a tiling, as well as a suitable notion of equivalence among different tilings. Delaney-Dress tiling theory provides a systematic approach to the complete enumeration of combinatorial equivalence classes of tilings in simply connected spaces. Computer implementations of algorithms based on Delaney-Dress tiling theory can exhaustively enumerate the combinatorial types of equivariant tilings in space forms [27]. This gives us a description of all combinatorially distinct tilings of an orbifold. For our purposes, we require an understanding of the distinct ways in which this combinatorial structure can be embedded on the orbifold, which in turn represent isotopically distinct tilings of the hyperbolic plane. For example, the Stellate orbifolds 2223 and 2224 can be decorated by a simple combinatorial structure, however this simple structure can manifest as an infinite set of isotopically distinct embedded hyperbolic tilings [18,19,48,47].
The classification of embedded combinatorial structures is precisely what this paper will address. We will generalize the Delaney-Dress combinatorial tiling theory to classify all isotopically distinct equivariant tilings of any hyperbolic surface of finite genus, possibly nonorientable, with boundary, and punctured. We consider here the 2-dimensional case, however, the related classifications for higher dimensional hyperbolic orbifolds is also briefly discussed. Our approach is constructive and therefore allows, in theory, a complete enumeration of such classes of tilings. These results will facilitate the future development of the EPINET database (Euclidean patterns in non-Euclidean tilings) [1], the central resource of the enumeration of Euclidean structure through hyperbolic tilings.
Throughout this paper, we make heavy use of the notion of orbifolds [57] and mapping class groups [20]. The connection of equivariant isotopic tiling theory and mapping class groups is novel, however, there is a well-known con-nection between the Teichmuller space of Riemann surfaces of genus g and certain tilings of the hyperbolic plane with 4g sided geodesical polygons that we will use as inspiration [20]. We will also derive some algorithms to enumerate all equivariant tilings on a hyperbolic Riemann surface in its uniformized metric. Note that this also produces tilings for other Riemannian surfaces by uniformizing the metric within its conformal equivalence class.
This paper is structured into five main sections which cumulatively build the connection between equivariant isotopic tiling theory and mapping class groups. We clarify several smaller questions along the way, building intuition of previous results in a new context. We begin with section 2 (Symmetry Groups of Tilings and Orbifolds), where we recapitulate the notion of two-dimensional developable orbifolds and expand the framework to incorporate more general classes of orbifolds with punctures and boundary. This is followed by section 3 (Isotopic Tiling Theory) where we generalize combinatorial Delaney-Dress tiling theory to encode isotopically distinct tilings of surfaces in terms of generators of the symmetry group. In section 4 (Outer Automorphisms) we will elucidate the connection between outer automorphisms and the generators that encode isotopically distinct tilings. Then, having laid all the groundwork, we will introduce the mapping class group (MCG) of orbifolds in section 5 and prove fundamental results facilitating its applications to tiling theory. In section 6, we establish relations between the spaces of tilings of covering spaces. This paper represents the theoretical foundation for an enumeration of isotopy classes of tilings on surfaces. The implementation of these results will appear in [37].

Symmetry Groups of Tilings and Orbifolds
We begin with orbifolds [12,57,3]. Let X be a 2D space form, i.e. a simply connected Riemannian manifold X with constant sectional curvature. We only work with developable orbifolds, which means that the orbifold O is given by X /Γ , where Γ ⊂ Iso(X ) is a discrete subgroup. The difference between X /Γ as a topological space and as an orbifold is that for the orbifold structure, one retains the information concerning Γ and can reconstruct the topological space X from X /Γ and vice versa. The group Γ is called the fundamental group of the orbifold O. In the classical orbifold setting, Γ is required to act cocompactly. We will only require the codomain to have finite area in its uniformized metric, i.e. the metric induced by X .
In particular, we are interested in the case X = H 2 , where Γ is a NEC group (non-Euclidean crystallographic group), or a hyperbolic orbifold group. The hyperbolic case is also the only case that admits infinitely many isomorphism types of hyperbolic groups Γ , i.e. there are infinitely many non-diffeomorphic hyperbolic orbifolds.
Let O be a 2D orbifold. We can identify the symmetry groups using Conway's orbifold symbol, as described below, but extended by generators for the non-classical features our orbifolds might have, i.e. hyperbolic transfor-mations H i of H 2 , corresponding to non-mirror boundary components of O and parabolic transformations P j corresponding to punctures. The diffeomorphic structure of O is determined by the Conway symbol for its fundamental group Γ := A · · · H i · · · P j · · · abc · · · × · · · • · · · . There are generators for the translations associated to each handle, given by X and Y , and going around a handle in an oriented way corresponds to the curve associated to the commu- There are also generators for each gyration point of order A, and for a curve γ going around the gyration point once we have γ A = 1, where we interpret the curve as a deck transformation [52]. For each mirror we have the usual Coxeter group relations, which depend on the angles of the intersecting mirrors. However, somewhat more subtle, in the case where the interior of the orbifold contains nontrivial features, we actually need to choose one mirror per mirror boundary component that we give two generators P and Q, ordered in positive orientation corresponding to its two mirror halves and one generator λ for the curve that goes around this boundary component once in positive orientation. We then add the relation P = λ −1 Qλ. Next, going around a cross-cap corresponds to a generator ω with Z 2 = ω, where Z corresponds to the curve entering the cross-cap once. There is one global relation for an orbifold, namely, the product of all Greek letters (plus the nonclassical elements) in the above has to be trivial, i.e.
We shall refer to this presentation as the standard presentation of the fundamental group of O. To standardize notation, we can also assume that in the presence of a crosscap, all handles are replaced by two crosscaps each [22]. We will only consider the geometric generators of the above form for orbifold groups in this paper. Note that there is a description of the deck transformations in Γ as homotopy classes of curves on the orbifold O [52,12], which is important to us.
The elements of an orbifold fundamental group Γ can be assigned types according to their algebraic properties and their action on the hyperbolic plane. Similar to [42], we define the type of an element in Γ as follows. Torsion elements that preserve the orientation of H 2 are the elliptic transformations of a given order. Mirrors represent torsion elements of order 2 that reverse the orientation. Orbifold groups like Γ , when viewed as conformal transformations of the upper half plane U ⊂ C, have an associated limit set Λ ⊂ R. The complement of Λ in R, C, has more than one connected component if O has a boundary. The conjugates of powers of the hyperbolic transformations associated to the boundary components of O map some component of C to itself and are called boundary hyperbolic. Also, there are the parabolic transformations corresponding to the punctures and the orientation preserving as well as the reversing hyperbolic transformations associated to the genus of a surface. All of these designations give the type of an element in Γ. We call an automorphism of Γ type preserving if it preserves the types of all elements in Γ.

Isotopic Tiling Theory
Tesselations of H 2 can be described using combinatorial tiling theory [14,27]. Combinatorial tiling theory classifies all possible equivariant combinatorial types of tilings on space forms X . It deals with the case that each tile is a closed and bounded disc and the symmetry group of the tiling acts cocompactly. A set T of such topological discs in X is called a tiling if every point x ∈ X belongs to some disc (tile) T ∈ T and if for every two tiles T 1 and T 2 of T , T 0 1 ∩ T 0 2 = ∅, where S 0 denotes the interior of a set S. All tilings in this paper will be assumed to be locally finite, meaning that any compact disc in X meets only a finite number of tiles. More specifically, combinatorial tiling theory is a barycentric subdivision of the tiles and subsequent tracking of the combinatorics of the resulting chamber system to give the coordination sequence of the tiling.
We call a point that is contained in at least 3 tiles a vertex, and the closures of connected components of the boundary of a tile with the vertices removed edges. The only exception to this are two-fold rotational centers of symmetry, which we will also consider to be vertices. Let T be a tiling of X and Γ be a discrete subgroup of Iso(X ). If T = γT := {γT |t ∈ T } for all γ ∈ Γ , then we call the pair (T , Γ ) an equivariant tiling and Γ its symmetry group. We call two tiles T 1 , T 2 ∈ T equivalent or symmetry-related if there exists γ ∈ Γ s.t. γT 1 = T 2 . We call the subgroup of Γ that leaves invariant a particular tile T ∈ T the stabilizer subgroup Γ T . A tile is called fundamental if Γ T is trivial and we call the whole tiling fundamental if this is true for all tiles. An equivariant tiling is called tile-, edge-, or vertex-k-transitive, if the number of equivalence classes under the action of the symmetry group is k. Note that the above definitions do not require Γ to be the maximal symmetry group for the tiling T . A fundamental tile-1-transitive equivariant tiling (fundamental tiling for short) (T , Γ ) has a single type of tile that is a fundamental domain for Γ and any fundamental domain for Γ also gives rise to such a tiling.
The above represents the framework for combinatorial tiling theory as outlined in [27]. We will generalize the theory to work with more general symmetry groups and also non-simply connected spaces. We will now detail a slightly different point of view using orbifolds and a concrete realization of a symmetry group Γ in Iso(X ), where we particularly emphasize the case X = H 2 .
One can view tilings as combinatorial structures or classes of decorations on orbifolds. This is based on the simple observation that any tesselation has the symmetry group of a developable orbifold. The underlying topological space of O can be extracted from any fundamental domain for Γ in H 2 , with appropriate edge identifications corresponding to the action of the generators of Γ on the fundamental domains boundary. The action of Γ also gives rise to a fundamental transitive tiling. Each fundamental transitive tile can also be interpreted as a (bordered) fundamental domain and can thus each be seen as a possible canvas, on which we can draw any orbifold decoration (after getting rid of the boundary edges, if necessary). Note that the tiling that results from the drawing can exhibit a higher degree of symmetry than what the symmetry group of the equivariant tiling suggests.
Geometrically, we first identify a set C of curves such that cutting O along C produces a fundamental domain, with the edge identifications (from Poincaré's theorem) yielding the given presentation of Γ [40,39]. For the decorations, one now draws arbitrary curves on O. Then, cutting open O along C and imposing Γ produces any tiling with symmetry group Γ . However, not all decorations produce an equivariant tiling by closed discs. In the language of Delaney-Dress tiling theory, each chamber system encoding an equivariant tiling with symmetry group Γ essentially corresponds to a triangulation of O = H 2 /Γ. 1 When viewing tilings as combinatorial decorations on orbifolds, it becomes natural to consider the more general situation of finite volume orbifolds and thus more general symmetry groups for the tilings than for classical Delaney-Dress tiling theory. There are a number of ways of approaching this problem. In [14], the statements made for the case of bounded tiles work just as well for an equivariant tiling theory for symmetry groups with cusps 2 . For the theory to incorporate punctures, one still looks at tesselations in terms of chamber systems and geometric cell complexes and treats the cusps as marked points belonging to the surface, but when embedding these into a manifold to obtain a tesselation, one needs to remove the cusps before embedding. Geometrically, the idea corresponds to pushing the punctures to the boundary of the unit circle in the Poincaré model for H 2 . Alternatively, a puncture in the orbifold, corresponding to a parabolic transformation in H 2 , can be seen as the limit of a sequence of gyration points of increasing order, with order ∞. This is in line with the Conway notation for orbifolds. From this point of view, the tilings for finite volume orbifolds with punctures are attained as limits of tilings for orbifolds where the puncture is a gyration point of increasing order. We will treat the cusped case in the same way as the classical case, except where we highlight differences, but with order ∞ singular points.
The theory for orbifolds with boundary can be dealt in the following way. We can replace all boundaries with mirrors, with each mirror/boundary component contributing a −1 to the Euler characteristic. The resulting orbifold induces a tesselation of H 2 . We now apply combinatorial tiling theory as before, and simply remove the mirrors, replacing the mirror symmetries with the hyperbolic transformations that represent the boundary deck transformations in Iso(H 2 ). The boundary of the tesselated submanifold of H 2 is then a result of the boundary and symmetry operations of the original smaller orbifold it covers.
For simply connected spaces and combinatorial classes of tilings, starting from the fundamental tilings, all other equivariant tilings with the same sym-metry group are obtained by using GLUE and SPLIT operations [4,27]. The different combinatorial types of a fundamental domain for a given classical orbifold were classified in [40]. Using the Delaney-Dress (D)-symbol, one can give unique names to the combinatorial structures on 2-orbifolds that represent tilings on space forms, which can be used for enumeration purposes [13]. We will subsequently focus on fundamental tilings. Given the generators, described in section 2, of a symmetry group G ⊂ Iso(H 2 ), the D symbol describes how the group acts on the associated chamber system of a tiling [14], where the chambers are triangles in a triangulation of the orbifold. The tiling itself is obtained from a decorated fundamental domain for G, with the generators acting on its boundary edges (which are not necessarily a part of the tiling). By the Poincaré theorem, a set of (geometric) generators of G all map part of the fundamental tile's boundary to itself to yield a presentation of the symmetry group, and the D symbol tells us in which way. Even if we restrict to geodesically bordered tiles, it is not the case that the D symbol uniquely defines a tiling up to isometries, as it does not even unambiguously define a metric on G's associated orbifold O. This is also true if one fixes a particular set of generators for G acting on the fundamental tile.
For example, what can happen is that the fundamental tile has a vertex that is not situated at a point of increased symmetry or that two metrically distinct Dirichlet fundamental domains yield the same combinatorial tiling structure.
The Teichmueller space T (G) is the space of type preserving, discrete faithful representations in PGL(2, R) (for orientable orbifold groups, one usually restricts to PSL(2, R)) of the abstract hyperbolic group G with standard presentation, modulo conjugation by elements in PGL(2, R). This space carries a natural topology, namely the subspace and subsequent quotient topology of Hom(G, PGL(2, R)), which itself is endowed with the compact-open topology. The topology of G is the discrete one and PGL(2, R) carries the topology it inherits from its usual structure as a Lie group. It is well-known [57] that T (G) is (component-wise) homeomorphic to R k for some k. The generalization to our more general orbifolds just means that we expand, by ideal hyperbolic triangles, the collection of primitive orbifolds with unique hyperbolic structures that assemble to produce more complicated orbifolds and continue with the same arguments presented in [57]. This implies that two different sets of generators for G in Iso(H 2 ) can be continuously deformed into one another in H 2 . The small caveat here is that for orientable G, there are two representations with opposite orientation (see section 4) in PGL(2, R), so the connectedness of T (G) is only true for representations of the same orientation.
The importance of the above is the observation that during the process of continuously deforming one representation of G into another, the combinatoric structure of the chamber system associated to the tiling remains invariant. Therefore, perhaps somewhat surprisingly, there is a set of combinatorial instructions for how to decorate the fundamental domain to produce a particular tiling from the generators is independent of the particular representation of G in PGL(2, R). This set of instructions can be read off the D symbol, see figure 1 for an example. Within a combinatorial class of fundamental tiles, we can interpret the other fundamental tiles with different positions for the generators as obtained by shearing the original one. This deformation can in fact be realized by a quasi-conformal mapping and induces a metric on the orbifold. In particular, one can calculate the metric properties (diameter,...) of a given orbifold realization, properties which are potentially of interest to the natural sciences.
We will now turn to an important example that we will use to highlight the general situation. Consider the fundamental 4g polygon of a closed hyperbolic Riemann surface S of genus g with given hyperbolic metric. The construction of a tiling starts from a given point x ∈ S which is the base point of the generating curves {γ i } 2g i=1 for the fundamental group of S. Within each homotopy classes for the curves γ i , there is a geodesical representative. Cutting the surface along these geodesics produces a hyperbolic tile and tesselation. The combinatorial structure of the associated hyperbolic tiling is uniquely determined, as there are 4g copies of the fundamental tile around every vertex, which does not depend on how the edge identifications are realized, i.e. on the presentation of π 1 (S). In particular, there is only one associated combinatorial class of fundamental tiling. On the other hand, the choice of base point for the construction of a fundamental domain for the generators produces a plethora of metrically distinct fundamental tilings, which are all derived from the same point in T (G). What different types of fundamental tilings can we create in this way? Any other tiling starts from a different point p ∈ S, and there is a path c connecting x and p. The path can be extended to an isotopy of S. In this way, c uniquely determines a homeomorphism up to isotopies from (S, x) to (S, p), following results relating to the point-push map in [20]. In particular, if one fixes a reference set of generators of π 1 (S), the resulting isotopy extension only leaves the set of generators invariant on S if the path c induces a trivial homeomorphism up to isotopy. This means that if we fix what the generators {γ i } map to in Iso(H 2 ), there is only one isotopy class of tilings associated to S and {γ i }. The combinatorial information needed to produce the corresponding tiling is then simply given by any base point needed to construct the associated fundamental domain. The case p = x is of particular interest. For nontrivial curves c, the generators in Iso(H 2 ) change, but because the induced automorphism on π 1 (S) by any such curve c is always inner, by the Dehn-Nielsen-Baer theorem [20], it corresponds to the isotopically trivial homeomorphism of the surface and does not change the tiling.
The same line of reasoning works for more general orbifolds O and their fundamental groups G, possibly with repeated use of the above arguments for more than one randomly chosen point, and explains why within a fixed set of generators for π 1 (O) ⊂ Iso(H 2 ) and combinatorial type of fundamental tiling with the generators acting on its boundary, the isotopy type of decoration does not depend on the choice of random points required in the construction. Note that this does not preclude different sets of generators with the same combinatorial decorations to yield the same isotopy class of fundamental tiling. A class of examples is discussed in section 7 below. A related idea to produce all combinatorial types of fundamental domains is to randomly choose points  of H 2 w.r.t. which the Dirichlet fundamental domains eventually produce all combinatorial types of fundamental tilings for any given group G (by theorem 3.3 of [40]). However, the construction of the Dirichlet fundamental domain is insensitive to the generators chosen for G. Generally, the edges of the fundamental tile can be given purely in terms of the generators, as edges connecting symmetry points, or randomly chosen points. The random points show up in the triangulation that is the chamber system of the orbifold when a vertex is not located at an increased symmetry site. This situation can be read off the D symbol. Using this approach to fundamental tilings from the chamber system related to the D-symbols gives a completely algebraic/combinatorial way of producing the fundamental tilings from the generators of G. In practice, this invariant description in terms of generators comes from simply producing a combinatorial version of a tiling from the D-symbol and then placing the vertices in the associated decoration accordingly, see figure 1. In doing so, the vertices have to be given in terms of their positions relative to the generators. 3 As an illustration (figure 1), consider the hyperbolic orbifold group G = 2 1 2 2 2 3 4 4 , where the subscripts track the positions of the generators. The placements of the generators in H 2 in figure 1a allows a fundamental tiling for the supergroup 2224 simply by considering the convex hull of the indicated points. Now, there are two ways that 2224 sits inside 2224. One is obtained by reducing the symmetry of the 4-fold rotation point, corresponding to figure 1a. The other is obtained by doing the same to a 2-fold rotation center, as has been done in figure 1b. This reflects the fact that there are only two combinatorial classes of fundamental tilings in H 2 for 2224. By the above dis-cussion, we can combinatorially give a description of the edges belonging to the fundamental tiling. In figure 1a, consider the rotations corresponding to the generators r 1 , ..., r 4 , with centers c 1 , ..., c 4 . Because the tiling is obtained by doubling the fundamental tiling of 2224, it is straightforward to see that the corners/increased symmetry points on the polygon's boundary correspond clockwise, starting at c 1 , to the points c 1 , c 2 , c 3 , c 4 , r 4 (c 3 ), r 4 r 3 r −1 4 (c 4 ). This procedure readily generalizes to arbitrary stellate orbifolds, i.e. those with only rotations for generators. Given any generators r 1 , ..., r 4 in Iso(H 2 ), this description of edges defines a fundamental tiling, in this case with totally geodesic edges, regardless of the generators placement in H 2 . Similarly, for the fundamental tiling of figure 1b, the edges are given by hyperbolic lines connecting the points c 1 , Figure 1c illustrates that this relation for the edges still holds and we obtain a fundamental tiling with symmetry group G, this time without the additional symmetries that we used to find the end points of the edges of the fundamental tile on whose boundary the generators are positioned. This example illustrates that it can be very helpful to look at versions of the equivariant tiling in question that exhibit more symmetries than the given symmetry group.
In general, we want to classify equivariant tilings of a hyperbolic Riemann surface S in its uniformized metric, i.e. given a fundamental hyperbolic polygon of S in H 2 , we want to find all ways of equivariantly tiling it, with fixed symmetry group G ⊂ Iso(S) ⊂ Iso(H 2 ). We consider equivariant tilings with the same symmetry group that are isotopic in S equivalent. This is somewhat different to the situation of Delaney-Dress tiling theory, where combinatorially equivalent tilings of H 2 are identified even if they are not isotopic. This discrepancy is essentially due to the different representations of G being connected by paths in T (G).
As far as isotopic tiling theory is concerned, it is not enough to consider just the abstract group G and the associated D symbols in our more general setting. Instead, it is important to use the method of producing fundamental tilings from D symbols along with specific generators for G as outlined above. There is a way to carefully choose only those sets of locations for generators for G that yield a priori different fundamental tilings of S (see sections 4 and 6 below). We will see in section 6 that these different sets of locations for generators for G give rise to non-isotopic tilings of S, for any decoration that is sufficiently complicated. It is a well-known fact that any closed curve on S has a unique geodesic representative in its isotopy class, so we will only work with piece-wise geodesic decorations, with breaks occuring only at vertices.
Consider tiling the genus 3 fundamental polygon of the Riemann surface S in H 2 with symmetry group 246. There are three different versions of the 22222 subgroup that are supergroups of π 1 (S). In hyperbolic geometry, by Hurwitz' theorem, there is a smallest (area-wise) possible hyperbolic group G 0 that is a supergroup of π 1 (S) and all three versions of 22222 will be a subgroup of G 0 and we see that 246 = G 0 . Each version of 22222 now has to be treated independently of the others when classifying all isotopy classes of equivariant tilings on S. Indeed, the fundamental tilings for every possible set of generators for each of these groups are non-isotopic as tilings on S (see section 6 below).
Before we go on to introduce new tools for tackling the new challenge of finding appropriate sets of generators for the symmetry groups of tilings, we would like to point out that the GLUE and SPLIT operations' validity remains unchanged in this new setting. One could ask if two different sets of generators S 1 , S 2 for the same group that produce different fundamental tilings lead to the same tiling of S after a sequence of such operations. If this were the case, then firstly the sequence of operations would be different. However, this would mean that these two different sequences of operations, each applied only to tilings derived from S 1 would yield combinatorially equivalent tilings. These are equivalent in the classical Delaney-Dress tilings theory, so no additional ambiguity emerges by us distinguishing between tilings associated to different sets of generators for the symmetry groups. What this statement expresses is that it is, in a way, very natural to consider the isotopy classes of tilings w.r.t. a set of generators for the symmetry group. Furthermore, this result is very important for enumerative to isotopic tiling theory.

The Group of Outer Automorphisms
Let G be a group. The group of all automorphisms of G is denoted by Aut(G). Conjugation by any element g ∈ G induces an automorphism c g (g) := ggg −1 forg ∈ G . Such automorphisms are traditionally called inner automorphisms, and the normal subgroup of all of them is denoted by Inn(G) := {c g |g ∈ G}. The picture is easiest to understand and yet very general in the case where G is a group of automorphisms of a space in some category. Consider now the case where G is a hyperbolic orbifold group. In its universal covering space H 2 , one finds special points of increased symmetry, i.e. invariant subsets, corresponding to a geometric realisation of the elements of G. In some cases, these points correspond to a singleton that is fixed under some element of G, while in other cases, they are submanifolds that are left invariant by an element of G. By the location of generators of G, we really mean the locations in H 2 of the invariant subsets. We fix a set of generators G 1 ⊂ Iso(X ) for G and consider subsets S of the elements of G ⊂ Iso(X ) with |S| = rank(G). We are interested in the following question: When do we have < S >= G? Interpreted within the context of group automorphisms starting from G 1 , which will correspond to the identity morphism, we see that they are exactly those subsets S that correspond to an element of Aut(G). Note, however, that we are not interested in the full group of automorphisms.
Instead, we will restrict our attention to the subgroup of type preserving automorphisms. This restriction is exactly what is needed to ensure that the combinatorics of general tilings are invariant when given as decorations of the associated orbifold (theorem 2).
Any tesselation with symmetry group G is clearly invariant under an inner automorphism of G. The converse is also true -the inner automorphisms of G are the only orientable automorphisms that leave invariant any decoration of fundamental domains for compact orbifold groups G. We prove a version of this statement in theorem 2. One way to think about this is to look at the relation between orbifold group elements and curves on the orbifold, which in turn can be interpreted as decorations lifted to the universal cover. Thus, when a sufficiently complicated decoration of the fundamental domain is invariant w.r.t. an (orientation preserving) homeomorphism of the underlying orbifold, the underlying homeomorphism must be isotopically trivial because it fixes all curves and therefore orbifold elements. This means that it corresponds to an inner automorphism of G by theorem 2 below. In case of noncompact orbifolds, this statement is only true for geometric automorphisms. A geometric automorphism is one that is realized by a homeomorphism of H 2 , see definition 1. A nongeometric automorphism can change the combinatorial structure of the decoration of the orbifold.
We have now further reduced our original surface decorating problem to the study of the group of outer automorphisms of a hyperbolic orbifold symmetry group. However, we are not interested in the full group of outer automorphisms, because in the general case of orbifolds with boundaries or punctures, the designation of the type of the generator as a hyperbolic translation or a boundary parabolic transformation is important to us. When decorating an orbifold, thereby producing tesselations, we want the number and different types of generators to be fixed so as to preserve the original combinatorial structure of the tesselation. Another important point is that changing the orientation of an orientable decorated orbifold does not impact the locations of the decorations on the orbifold and therefore the associated tesselation remains invariant. Therefore, we are also only interested in a representative automorphism out of the class of orientation preserving or reversing geometric automorphisms. While orientation is a geometric notion, there is an algebraic analogue [58] that captures the intuition of the geometric notion, so it makes sense to talk about the orientation of automorphisms of abstractly defined groups.
We are now prepared to formulate a result that highlights the importance of the 2D setting. The Mostow rigidity theorem implies that the deformation space of finite volume hyperbolic structures on an orbifold of dimension ≥ 3 is a singleton. In particular, Out(O) is trivial and once we have chosen generators for the symmetry group, there is no way to obtain other generating sets via a geometric automorphism. In effect, this means that the combinatorial tiling theory for such non-simply connected hyperbolic manifolds is the same as classical combinatorial tiling theory, which does not take into account different sets of generators and all possible isotopy classes of tilings can be attained by randomly choosing points w.r.t. which one produces the Dirichlet fundamental domain.

The Mapping Class Group of an Orbifold
Our goal is to classify all of the relevant locations for generators for hyperbolic orbifold groups in Iso(H 2 ) that lead to different tilings when decorated in an invariant way, according to D-symbols as outlined in section 3. Having laid all the groundwork, we now introduce the mapping class group (MCG) of orbifolds and prove fundamental results facilitating its applications to tiling theory.
Let O be a not necessarily orientable compact hyperbolic 2-orbifold, possibly with finitely many punctures and some boundary components. Denote by O its underlying topological surface with weighted marked points at conical singularities of order equal to the assigned weight. Punctures can be treated  [15], whose results are also proved for nonorientable surfaces and not necessarily hyperbolic ones. Now, all ordinary boundary components of O are disjoint from labelled ones representing mirrors. Therefore, mirrors are treated in the same way as boundary components but disjointly and the proof remains correct word for word. Thus, we have

Lemma 1 Let [f ], [g] ∈ Mod(O). Then [f ] = [g]
if and only if f and g are homotopic in O 0 .
where Σ, as above, denotes the singular locus of O. Then p : Z → O 0 is a non-branched and regular cover of connected topological spaces. Furthermore, π 1 (O 0 ) has generators X i corresponding to curves around the isolated points of the singular locus. In particular, π 1 (O 0 ) has the same number of generators as π 1 (O) in its standard presentation if one excludes the mirror symmetries from the latter. We now chose appropriate base points z 0 and x 0 for Z and O 0 such that p(z 0 ) = x 0 , so we can talk about concrete subgroups of the fundamental groups involved.
The groups of deck transformationsΓ of the cover p : Z → O 0 and π 1 (O 0 ) are related byΓ = π 1 (O 0 )/π 1 (Z). Here, we interpret π 1 (Z) as a subgroup of π 1 (O 0 ) in the usual way, i.e. as the push forward of p as p (π 1 (Z)). Clearly, π 1 (Z) equals the normal closure of the elements X oi i , where the o i are the orders of the X i in O, since these are exactly the relations imposed on the generators of π 1 (Z) when passing over to π 1 (O 0 ) with p. Let f ∈ Hom(O), then, by construction, f : O 0 → O 0 preserves the order of branching of p. We therefore have that f (X oi i ) ∈ π 1 (Z). This is exactly the criterion (see [26, prop. 1.33]) for the map f • p to lift to a map f 1 : Z → Z.
We will check that f 1 can be uniquely extended to the closure of Z in H 2 and then, if necessary by reflections, to a map f * on all of H 2 , following the arguments in [42, p. 500]. Take a small neighborhood U of one of the punctures in Z. Then f 1 (U ) has infinite cyclic fundamental group, meaning it is either a punctured disc or an annulus. The case of the annulus is easily excluded, which means that f 1 permutes the punctures and can be extended to the closure of Z. The extension is unique and the only ambiguity here stems from lifting f to f 1 , but two such lifts are related by a deck transformation in Γ. We obtain an automorphism α of Γ by γ → f * γ(f * ) −1 , where the group relations are easily checked.
Below we will need the following theorem, which is proved in [43] for orientable case.
Theorem 1 Suppose f ∈ Hom(O) and let f * be its lift like above. Then the induced automorphism α is the identity automorphism of π 1 (O) if and only if f is isotopic in O 0 to the identity mapping.
We will see below in the proof of theorem 2 that homotopic mappings in O 0 yield the same automorphism of Γ , which deals with one direction. The proof of the other direction requires careful study of the proof in [43]. Indeed, the proof works in the exactly the same way as presented there, but we need to exchange one of the key ingredients. The following lemma replaces lemma 1 in [43].
Lemma 2 Suppose S is a surface like above (possibly obtained, like O 0 , from a surface with features O) and g is a homeomorphism of S. If there exists an arc c from a point 0 ∈ S to g(0) such that α is homotopic to cg(α)c −1 for all simple closed curves α based at 0 that are disjoint from the boundary, then g is homotopic in S to the identity.
In [5], the corresponding statement for orientable, closed surfaces is proved using presentations of the surface in H 2 . All of the arguments used there also work for the surfaces with features that we study, as long as we keep in mind the following. First, the orbifold fundamental group has a natural interpretation in terms of orbifold loops. Second, the construction of the lift of a map f in [5, p. 20] has to be replaced by the construction of f * given above. Lemma 1 then yields the isotopy of the theorem. Lastly, recall that the fundamental group π 1 (S) of a surface is generated by simple closed curves, also in the orbifold case.
The MCG of a space is often studied by looking at the action of the homeomorphism classes on isotopy classes of curves. For example, let O = 2222a, with a ≥ 2, then Mod(O) is one of two different types of groups. If a = 2, Mod(O) = Mod(S 5 ), the usual MCG of the 5-punctured sphere with punctures p 1 , ..., p 5 corresponding to the hyperbolic rotations r 1 , ..., r 5 . If a > 2, Mod(O) is the subgroup of Mod(S 5 ) corresponding to those homeomorphism classes that fix the conical singularity a. It is well-known that the elements of finite order in π 1 (2222a) are characterisic, i.e. are preserved as a set under automorphisms. If, moreover, an automorphism α is type preserving and orientation preserving, α(r i ) = tr j t −1 . It is impossible that this kind of transformation sends an elliptic transformation to a nontrivial power of itself. Indeed, assume that r d = trt −1 for some d > 1. Then r d−1 = trt −1 r −1 = [t, r] is elliptic. However, by [23, pp. 191-193], the commutator of an elliptic transformation with any other transformation cannot be elliptic. Together with the general statements in theorem 2 below, the above reasoning implies that these types of transformations never yield an automorphism of the whole orbifold group, even if they do yield ones of the local group. This generalizes an observation made in [18] and [19], and discussed in more detail in [16], whereby the placement of generators in certain domains of H 2 is prohibited.
Any automorphism of π 1 (O) with conical singularities can be assigned an orientation with the expected property that all orientation-preserving automorphisms form a subgroup of index 2 in all automorphisms [58]. In addition, type and orientation-preserving automorphisms map the elliptic generators to conjugates of elliptic generators of the same order [58]. We denote with Out + (π 1 (O)) the subgroup of orientation and type preserving automorphisms, which contains all inner automorphisms. The well-known Dehn-Nielsen-Baer theorem can be generalized to Mod(O) ∼ = Out + (π 1 (O)) [42]. We will prove the following, in much the same way, by providing an explicit isomorphism.
Theorem 2 Let O be a nonorientable hyperbolic orbifold. Then the MCG Mod(O) defined above is isomorphic to Out t (π 1 (O)), the group of type preserving outer automorphisms. If O is orientable, then the MCG Mod(O) is isomorphic to Out + (π 1 (O)), the group of orientation and type preserving automorphisms.
Proof We only need to worry about the nonorientable case. Define a morphism ϕ : Mod(O) → Out t (G) by ϕ(f )(γ) := f * γ(f * ) −1 for γ ∈ G, where f * is the lift of f defined above. Notice that the ambiguity off and therefore f * means that ϕ is only defined up to inner automorphisms. Moreover, two isotopic maps in Hom(O) yield the same image in Out t (G), so ϕ is well-defined on isotopy classes. Indeed, we can assume w.l.o.g. that f is isotopic to the identity and fixes some base point x 0 ∈ O 0 , and repeat the arguments found in the proof of theorem 1 in [42], with the addition of the arguments found in [58, p. 152] to conclude that we also obtain the orientation-reversing automorphisms of G by the orientation-reversing lift of an orientation-reversing homeomorphism f .
In [41,Theorem 3], it is proved that any automorphism of a hyperbolic orbifold group with compact codomain is realized geometrically, i.e. induced by a homeomorphism of H 2 . The proof there can be extended to finite area orbifolds using the uniqueness and existence of an extremal quasi-conformal mapping within an isotopy class of homeomorphisms of the hyperbolic plane as given in [2, p. 59, Theorem 2]. The only difference in the proof then is that instead of reducing to the case of a compact surface by passing over to a finite index subgroup, by the positive resolution of the Fenchel conjecture in [21,8,10], we pass over to the fundamental group of a possibly punctured and bordered orientable surface. This means that instead of every automorphism being realized geometrically as in the compact case, we obtain the statement that only the type preserving ones are realized, as this is the case for surfaces with boundaries and punctures. This last statement, instead of using the original Dehn-Nielsen-Baer theorem for compact surfaces, employs theorem 8.8 from [20] instead, which on account of us allowing homeomorphisms that are not the identity on the boundary holds for surfaces with boundary as well, as long as the automorphisms considered are type preserving.
We thus conclude that all type preserving automorphisms of G are realised geometrically and therefore ϕ is surjective.
For injectivity, assuming that f lifts to a homeomorphism f * that induces an inner automorphism, there is a lift of f that is the identity automorphism on G. Theorem 1 concludes the proof. 4 While theorem 2 is an important result, it is as of yet unclear how to use this isomorphism in general for practical purposes. The same proof holds in the Euclidean case, where the surjectivity of the homomorphism is true for the same basic reasons that it is true for hyperbolic orbifolds. 5 From the proof of theorem 2 and the fact that geometric automorphisms are type preserving, we also obtain the following.
Proposition 1 For a compact orbifold O, missing the boundary hyperbolic elements and punctures, every automorphism of π 1 (O) is realized geometrically, so the MCG Mod(O) is isomorphic to either the group of all outer automorphisms π 1 (O) or just the orientation preserving ones, depending on whether or not O is orientable.

Lifts of Mapping Class Groups
In an effort to relate the MCGs of some surfaces to the MCGs of covers of the surface, Birman-Hilden theory was introduced [6]. The idea is the following. Given a covering map p : S → X of surfaces, one may look at fiber-preserving homeomorphisms f : S → S that for all x ∈ X map the fibers p −1 (x) to p −1 (y) for some y ∈ X. If this is the case, then f induces a homeomorphism on X. Conversely, if a homeomorphism f on X lifts to a map and therefore a homeomorphismf on S,f must be fiber-preserving. If for any two fiberpreserving homeomorphisms on S that are homotopic as maps on S, there is a homotopy passing only through fiber-preserving homeomorphisms, then we say that p has the Birman-Hilden property. The importance of this notion is that the MCGs for surfaces are defined through homotopies and in order to relate the MCGs of both spaces, one needs to ensure that only isotopic homeomorphisms of X lift to isotopic homeomorphisms of S. It is known [59,Theorem 11.1] that if p is a finite-sheeted branched regular covering map of orbifolds, then p has the Birman-Hilden property.
As such, Birman-Hilden theory concerns itself with the well-definedness of a lift of an isotopy class of a map to an isotopy class in the covering space. However, this leaves open the question of the existence of a lift of a representative of an isotopy class of maps. We will also investigate the question of existence of lifts of homeomorphisms of orbifolds to their covering spaces. 6 Let p : O 1 → O be a covering map of orbifolds. Any hyperbolic orbifold (with the exception of non-developable orbifolds), O, possibly with punctures and non-empty boundary, can be presented as H 2 /Γ where Γ = π 1 (O) is a discrete subgroup of Iso(H 2 ). We have that H 2 → H 2 /Γ is a regular branched cover, where the branch locus is a (possibly non-discrete) nowhere dense set in O. Similarly, we have O 1 = H 2 /Γ 1 and we naturally have Γ 1 ⊂ Γ, with each of these groups acting as a group of deck transformations on the universal covering by H 2 . We are only interested in finite covers, which translates to Γ 1 having finite index in Γ , equal to the degree of p. For compact orientable surfaces, it is well-known that any finite index subgroup of the fundamental group is isomorphic to the fundamental group of a covering surface, whereas any infinite index subgroup is free.
We will start the subsequent discussion with results whose proofs do not, as far as we know, appear in the literature but can be carried out with well-known methods in the field. First we need the following definition, the notion of which was touched upon in section 4.

Definition 1
We call an automorphism α of an orbifold group Γ (with standard presentation) geometrical, if there exists a homeomorphism f of H 2 that is Γ fiber-preserving w.r.t. the universal covering of the orbifold by H 2 and induces α via α(γ) = f γf −1 , where γ ∈ Γ ⊂ Iso(H 2 ) if interpreted as a group of deck transformations on H 2 . This is equivalent to saying that f Γ f −1 = Γ, which again is equivalent to f inducing a homeomorphism of the orbifold H 2 /Γ. Vice versa, any homeomorphism f of a surface H 2 /Γ lifts to a homeomorphismf of the universal covering space p : H 2 → H 2 /Γ that is Γ fiber preserving by applying the lifting criteria in [26, prop. 1.33] and uniqueness of lifts to the map f • p and f −1 • p (possibly with continuous extensions for the branch points). A similar condition also holds for coverings by orbifolds, as the singular locus can be ignored for the existence of a lift away from branch points, only to be reinstated later by Riemann's removable singularity theorem. Recall, also, that theorem 2 implies that geometric automorphisms are exactly those that are type preserving.
We now state a further corollary of theorem 2.
Corollary 1 A geometric automorphism of an orbifold group Γ that induces an automorphism on an orbifold subgroup S ⊂ Γ induces a geometrical automorphism on S.
For the following theorem we will mostly follow the proof of theorem 8.2 in [59], but produce a slightly stronger result.
Theorem 3 Let G be the symmetry group of a hyperbolic orbifold O and G 1 a subgroup of finite index that is not cyclic. Then a geometrical automorphism α of G 1 is induced by a G fiber-preserving homeomorphism in Hom(O) iff α is induced by an automorphismα of G.
Proof If α is induced by a G fiber-preserving homeomorphism f , then f induces a homeomorphism on the orbifold H 2 /G as well as, by assumption, on H 2 /G 1 , and thus induces an automorphismα of G that stabilizes G 1 in G, which proves one direction.
For the other direction, first consider the situation for the at most index 2 subgroupÑ ⊂ G that contains only orientation preserving elements. We further pass to a finite index normal subgroup N ⊂Ñ of G, which we can take to be the fundamental group of a possibly punctured and bordered orientable surface. Now let α be induced by a homeomorphism h of H 2 such that α(n) = h • n • h −1 ∀n ∈ N, which w.l.o.g. can be chosen to be the uniquely determined extremal quasi-conformal mapping of H 2 satisfying this relation. This statement remains true even for orientation reversing homeomorphisms of H 2 by the uniqueness of the extremal maps, since they can be given by a quasiconformal map composed with an orientation reversing isometry of H 2 , which leaves invariant the dilatation. Now define for arbitrary g ∈ G ϕ =α(g)hg −1 .
For n ∈ N we obtain The fourth equality uses g −1 ng ∈ N , since N is normal. Now, α(n) and n act as isometries on H 2 , hence leave the dilatation of ϕ invariant, so by the uniqueness of extremal maps we obtain ϕ = h and thusα(g) = h(g)h −1 . Since g was arbitrary, h preserves G-fibers. However, byα(g) = h • g • h −1 , this also shows that h preserves the fibers of any intermediate subgroup between N and G and, in particular, G 1 , which proves the theorem.
The next is a slight generalization of [42,Lemma 11].
Lemma 3 If some automorphism α of the hyperbolic orbifold group G induces an automorphism α| N of a noncyclic subgroup N ⊂ G, then there is only one extension of α| N to G, i.e. if α| N = id N then α = id G .
Recall that a characteristic subgroup C of G is a subgroup that is invariant under all automorphisms of G. This means that C ⊂ ϕ(C) ∀ϕ ∈ Aut(G) and thus also C ⊂ ϕ −1 (C), i.e. ϕ(C) ⊂ C so that any ϕ ∈ Aut(G) induces an element of Aut(C). It is well-known that every finitely presented group contains a finite index subgroup that is characteristic.
Proof Indeed, let g ∈ G. By passing to a finite index subgroup, we first assume that N is normal in G. Then for any n ∈ N we have gng −1 = α(gng −1 ) = α(g)nα(g) −1 , i.e. g −1' α(g) commutes with every element of N . The condition that N is not a cyclic subgroup of G now implies that g = α(g), because a nontrivial element in G commutes only with elements of a cyclic subgroup it is a part of [23]. Now let S ⊂ G be an arbitrary subgroup, which contains a characteristic subgroup C of G. Any automorphism of G induces one on C and, since C is normal in S and G, the extension to both of them is unique. So, α| C extends uniquely to α| S and to α. If there was some other extension of α| S to an automorphism of G, then this would contradict the uniqueness of the extension of α| C to α.

Lemma 3 remains valid within the class of geometrical automorphisms.
We now give a short proof of the Birman-Holden property for general orbifold groups, which is somewhat different than that in [59].
Proposition 2 Let S ⊂ G be a finite index subgroup of the hyperbolic orbifold group G that is not cyclic. If a G-fiber preserving homeomorphism ϕ of H 2 is S fiber isotopic to the identity, then ϕ is G-fiber isotopic to the identity.
Proof By assumption, ϕ induces an automorphism α of G, which induces id S on the subgroup S, so by lemma 3 α = id G , which by theorem 2 implies that ϕ is G fiber isotopic to the identity.
An important technical consequence of the Birman-Hilden property in terms of isotopic tiling theory is that two sufficiently complicated tilings that arise from decorations w.r.t. a non-conjugate pair of sets of generators are never isotopic in S, even if they are topologically not distinct, i.e. they are related by a homeomorphism of S.
As a result of the Birman-Hilden property, given a covering of orbifolds O 1 → O 2 , with groups G 1 and G 2 respectively, a homeomorphism class in O 2 lifts to one in O 1 , iff there is a lift of a representing homeomorphism. By theorem 3, this is equivalent to the corresponding automorphism of G 2 restricting to an automorphism of G 1 . In other words, a geometric automorphism of G 1 , induced by a map f such that Consider the subgroup L ⊂ Hom(O 2 ) of homeomorphisms of O 2 that lift to homeomorphisms on O 1 and set A := Hom(O 2 )/L. Two elements f, g ∈ A are equal iff f g −1 ∈ L. This implies that the induced automorphisms A f , A g of G 2 satisfy A f (G 1 ) = A g (G 1 ). Said in another way, there are as many equivalence classes in A as there are isomorphic versions G 1 in G 2 that get exchanged by automorphisms of G 2 . In this situation, A f (G 1 ) has the same index in G 2 for all f . Since G 2 is finitely generated, there are only a finite number of subgroups in G 2 of a given index, so we obtain the following.
Proposition 3 Given the situation of the last paragraph, there are at most finitely many homeomorphism classes with representatives that do not lift.
The contents of this section open up possible investigations into more refined questions relating to isotopic tiling theory on a Riemannian surface S. For example, lemma 3 tells us that elements of the MCG that are supported in a particular subsurface give rise to automorphisms that leave invariant a subset of the generators. It is well-known that the MCG of any surface has generators that are supported in subsurfaces. Furthermore, one is often in a situation where one is only interested in a subclass of all isotopy classes of tilings of a given surface, in which case theorem 3 facilitates the investigation. Proposition 3 tells us that from a perspective of tilings as graph embeddings, there is only a finite number of topologically distinct graph embeddings into S that can be produced by changing the isotopy class of the embedding of the graph into the orbifold. The results furthermore add to the duality of the description of the MCG as a group of geometric transformations and as a group of algebraic transformations. In particular, the following important related questions can be examined from an algebraic or a geometric point of view.
-Which isotopically distinct tilings with the same symmetry group G are related by a homeomorphism of S?
-How does an element of the MCG of an orbifold relate to the MCG of a covering orbifold?
Note that in most cases these questions do not have a generic answer and depend on the set up, i.e. the conformal structure on S and the tiling.

Summary and Implications for Applications
We have developed a classification of all isotopically distinct equivariant tilings of a hyperbolic surface S of finite genus, possibly nonorientable, with boundary, and punctured. First, we find the smallest (in terms of area) possible symmetry group of S, which corresponds to a symmetry group G 0 of the hyperbolic fundamental polygon belonging to S. This smallest symmetry group G 0 exists as a consequence of generalizations of the classical Hurwitz theorem [46]. There are finitely many possible symmetry groups G for tilings such that G 0 ⊂ G ⊂ π 1 (S). Given such a G, we choose a set of geometrical generators of cardinality rank(G). From these generators, we obtain a set of fundamental tilings with symmetry group G as a decoration of the associated orbifold O. The decoration is specified up to isotopy by a combinatorial description from the Delaney-Dress symbol of the tiling. The mapping class group Mod(O) of O naturally acts on the set of sets of generators of cardinality rank(G). Thus, starting from the classical Delaney-Dress symbol for the fundamental tiling with the starting set of generators, one obtains all other isotopically distinct fundamental equivariant tilings with symmetry group G by repeated applications of Mod(O). For each of the resulting fundamental tilings, we independently apply the GLUE and SPLIT operations exactly in the same way as in the classical setting to eventually produce all equivariant tilings with symmetry group G. One caveat here is that in some examples, for example when a fundamental tile for a group generated entirely by rotations of the same order is obtained by doubling a fundamental tile for the index 2 supergroup that is generated by reflections, it is possible to find two different sets of generators that are nonconjugate but act on the boundary of the same fundamental domain. In such situations, it is possible that the tiling associated to a decoration of the orbifold is unchanged by an element of the MCG. In case of a fundamental tiling, one only needs to retain one of the two versions of the tiling to produce all tilings associated to the symmetry group. While G 0 is the smallest symmetry group commensurate with S, this group depends entirely on the hyperbolic finite area metric on S. Without reference to any specific hyperbolic structure, there are many possible chains of subgroups that yield potential symmetry groups of S. For example, the group 2226 appears as the smallest fundamental domain of the H surface in [54]. However, this group does not appear at all as a symmetry group of the P surface in [53]. Both surfaces are of genus 3. Also, 246 has no hyperbolic supergroups, even though 237 is smaller.
Theorem 2 implies that the MCG does not depend on the orders of the torsion elements of the orbifold group and as a result, abstract results on MCGs are important for applications. In our definition of the MCG, where homeomorphisms are allowed to change the boundary, surfaces with boundary do not necessarily have torsion free MCGs, in contrast to the classical situation. An important technical aspect of the EPINET enumerative project is that many tilings of the hyperbolic surface S in question are related by isometries of the surface that lift to symmetries of R 3 . When producing nets in R 3 , one only wants to produce one representative of these. Finite order elements of Mod(O) necessarily act as isometries of H 2 , because their dilatation has to be equal to 1, since we can assume the homeomorphism of H 2 inducing the automorphism to be an extremal quasi-conformal mapping. This also shows that there exists a homeomorphism f in Hom(O) with the same order as its image in Mod(O). This partially settles the Nielsen realization problem for orbifold MCGs, see [20] for further details in the classical setting. Now, any such isometry h transforms the invariant point set of any g ∈ π 1 (O) to a similar one and therefore satisfies hgh −1 ∈ π 1 (O), so h ∈ N (π 1 (O)), the normalizer in Iso(H 2 ). Now, suppose that h ∈ Iso(H 2 ) acts trivially by conjugation on G. Then, it would have to fix all of the fixed points on the unit circle at infinity of the hyperbolic translations in G. Since every hyperbolic orbifold sits inside a classical surface of genus at least 2, there are two independent translations and therefore, h fixes 4 points on the unit circle and must be the identity, so N (G) injects into Aut(G), where G itself acts as inner automorphisms of G. Now, hGh −1 ⊂ G implies that h preserves G orbits as x ∼ y implies hy = hgx =ghx for some elements g,g ∈ G and therefore h induces a conformal automorphism of the orbifold H 2 /G. Conformality of map can be checked locally with the help of Riemann's removable singularity theorem. Clearly, G itself acts trivially on the space of its orbits and lifting arguments imply that we actually get an isomorphism of groups CAut(H 2 /G) ≡ N (G)/G, where CAut(H 2 /G) is the group conformal automorphisms. In particular, this means that the normalizer is discrete and that f can be interpreted as an element of a hyperbolic supergroup of G. Note, however, that the conformal structure of H 2 /G can give rise to different towers of supergroups of G, which is particularly important because in some cases, the MCG is generated by torsion elements. This is of considerable importance for 3D net enumeration, since the group of symmetries H of most of the genus 3 triply periodic minimal surfaces do not admit further hyperbolic supergroups. Therefore, any finite order element of the MCG corresponds to an element in H. In particular, if we choose the realization of π 1 (S) and thus S as a hyperbolic tile such that π 1 (S) H, then all finite order elements of the MCG lift to genuine symmetries of S, which means that when enumerating 3D nets, one wants to get rid of these elements.
For example, the finite orders of elements of the MCG M n of an n-times punctured sphere were studied in [25], [38]. The result is that m is the order of an element in the MCG if and only if m divides n, n − 1, or n − 2, and there is even an explicit form of representatives of conjugacy classes of all finite order elements using the standard representation for these groups from braid groups. Using these, one can enumerate the cosets giving rise to symmetry classes of isotopy classes of tilings with rotational symmetry on a given Riemmanian surface S. For example, the group 22222 has the MCG of a 5 punctured sphere. All finite order elements correspond to isometries of 246.
An essential ingredient in any enumeration of isotopy classes of tilings comes from the fact that our MCGs have solvable word problem, which one can prove for our more general surfaces in almost the same way as has been done in [20] for closed surfaces, using the Alexander method. This allows an unambiguous and complete enumeration of all isotopy classes of tilings on hyperbolic Riemann surfaces by an enumeraton of MCG elements, a project which we pursue in [37].