A note on properly discontinuous actions

We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.

To the memory of Sasha Anan'in

Introduction
This note is meant to clarify the relation between different commonly used definitions of proper discontinuity without the local compactness assumption for the underlying topological space.Much of the discussion applies to actions of nondiscrete locally compact Hausdorff topological groups, but, since my primary interest is geometric group theory, I will mostly work with discrete groups.All group actions are assumed to be continuous, in other words, for discrete groups, these are homomorphisms from abstract groups to groups of homeomorphisms of topological spaces.This combination of continuous and properly discontinuous, sadly, leads to the ugly terminology "a continuous properly discontinuous action."A better terminology might be that of a properly discrete action, since it refers to proper actions of discrete groups.
Throughout this note, I will be working only with topological spaces which are 1st countable, since spaces most common in metric geometry, geometric topology, algebraic topology and geometric group theory satisfy this property.One advantage of this assumption is that if (x n ) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {x n : n ∈ N} is compact, which is not true if we work with nets instead of sequences.However, I will try to avoid the local compactness assumption whenever possible, since many spaces appearing in metric geometry and geometric group theory (e.g.asymptotic cones) and algebraic topology (e.g.CW complexes) are not locally compact.(Recall that topological space X is locally compact if every point has a basis of topology consisting of relatively compact subsets.) In the last three sections of the note I discuss several concepts related to properly discontinuous actions.In Section 5 I discuss cocompactness of group actions.In Section 6 I discuss group-invariant metrics.In particular, under suitable assumptions I will prove existence of an invariant complete geodesic metric (Theorem 25).In Section 7 I discuss fundamental sets and regions.The main result of this section is Theorem 61 which uses Voronoi tessellations to establish existence of fundamental regions and domains for free properly discontinuous actions on proper geodesic metric spaces.A left continuous action of a topological group G on a topological space X is a continuous map λ : G × X → X satisfying 1. λ(1 G , x) = x for all x ∈ X.
From this, it follows that the map ρ : G → Homeo(X) ρ(g)(x) = λ(g, x), is a group homomorphism, where the group operation φψ on Homeo(X) is the composition φ • ψ.
If G is discrete, then every homomorphism G → Homeo(X) defines a left continuous action of G on X.
The shorthand for ρ(g)(x) is gx or g • x.Similarly, for a subset A ⊂ X, GA or G • A, denotes the orbit of A under the G-action: GA = g∈G gA.
The quotient space X/G (also frequently denoted G\X), of X by the G-action, is the set of Gorbits of points in X, equipped with the quotient topology: The elements of X/G are equivalence classes in X, where x ∼ y when Gx = Gy (equivalently, y ∈ Gx).
The stabilizer of a point x ∈ X under the G-action is the subgroup G x < G given by {g ∈ G : gx = x}.
An action of G on X is called free if G x = {1} for all x ∈ X.Assuming that X is Hausdorff, G x is closed in G for every x ∈ X. Example 1.An example of a left action of G is the action of G on itself via left multiplication: λ(g, h) = gh.
In this case, the common notation for ρ(g) is L g .This action is free.

Proper maps
Properness of certain maps is the most common form of defining proper discontinuity; sadly, there are two competing notions of properness in the literature.
A continuous map f : X → Y of topological spaces is proper in the sense of Bourbaki, or simply Bourbaki-proper (cf.[6, Ch.I, §10, Theorem 1]) if f is a closed map (images of closed subsets are closed) and point-preimages The advantage of the notion of Bourbaki-properness is that it applies in the case of Zariski topology, where spaces tend to be compact 1 (every subset of a finite-dimensional affine space is Zariski-compact) and, hence, the standard notion of properness is useless.
Since our goal is to trade local compactness for 1st countability, I will prove a lemma which appears as a corollary in [20]: and X, Y are Hausdorff and 1st countable, then f is Bourbakiproper.
Proof.We only have to verify that f is closed.Suppose that A ⊂ X is a closed subset.Since Y is 1st countable, it suffices to show that for each sequence ( Remark 3.This lemma still holds if one were to replace the assumption that X is 1st countable by surjectivity of f , see [20]. The converse (each Bourbaki-proper map is proper) is proven in [6, Ch.I, §10; Prop.6] without any restrictions on X, Y .Hence: Corollary 4. For maps between 1st countable Hausdorff spaces, Bourbaki-properness is equivalent to properness.

Proper discontinuity
Suppose that X is a 1st countable Hausdorff topological space, G a discrete group and G×X → X a (continuous) action.I use the notation 1 quasicompact in the Bourbaki terminology Given a group action G × X → X and two subsets A, B ⊂ X, the transporter subset (A|B) G is defined as (A|B) G := {g ∈ G : gA ∩ B = ∅}.Properness of group actions is (typically) stated using certain transporter sets.Definition 5. Two points x, y ∈ X are said to be G-dynamically related if there is a sequence g n → ∞ in G and a sequence x n → x in X such that g n x n → y.
A point x ∈ X is said to be a wandering point of the G-action if there is a neighborhood U of x such that (U |U ) G is finite.Lemma 6. Suppose that the action G × X → X is wandering at a point x ∈ X.Then the Gaction has a G-slice at x, i.e. a neighborhood W x ⊂ U which is G x -stable and for all g / ∈ G x , Then the intersection satisfies the property that (V |V ) G = G x .Lastly, take The next lemma is clear: Assuming that X is Hausdorff and 1st countable, the action G × X → X is wandering at x if and only if x is not dynamically related to itself.
Given a group action α : G × X → X, we have the natural map where id X : (g, x) → x.
Definition 8.An action α of a discrete group G on a topological space X is Bourbaki-proper if the map α is Bourbaki-proper.
Lemma 9.If the action α : G × X → X of a discrete group G on a Hausdorff topological space X is Bourbaki-proper, then the quotient space X/G is Hausdorff.
Proof.The quotient map X → X/G is an open map by the definition of the quotient topology on X/G.Since α is Bourbaki-proper, the image of the map α is closed in X × X.This image is the equivalence relation on X × X which use used to form the quotient X/G.Note that the equivalence of (1) and (5) in the following theorem is proven in [6, Ch.III, §4.4,Proposition 7] without any assumptions on X.
Theorem 11.Assuming that X is Hausdorff and 1st countable, the following are equivalent: (1) The action α : G × X → X is Bourbaki-proper.
(3) The action α : G × X → X is proper, i.e. the map α is proper.
(4) For every compact subset K ⊂ X, there exists an open neighborhood U of K such that card((U |U ) G ) < ∞. (5) For any pair of points x, y ∈ X there is a pair of neighborhoods U x , V x (of x, y respectively) such that card((U x |V y ) G )) < ∞. (6) There are no G-dynamically related points in X. (7) Assuming, that G is countable and X is completely metrizable2 : The G-stabilizer of every x ∈ X is finite and for any two points x ∈ X, y ∈ X−Gx, there exists a pair of neighborhoods U x , V y (of x, resp.y) such that ∀g ∈ G, gU x ∩ V y = ∅.(8) Assuming that X is a metric space and the action G × X → X is equicontinuous3 : There is no x ∈ X and a sequence h n → ∞ in G such that h n x → x. (9) Assuming that X is a metric space and the action G × X → X is equicontinuous: Every x ∈ X is a wandering point of the G-action.(10) Assuming that X is a CW complex and the action G × X → X is cellular: Every point of X is wandering.(11) Assuming that X is a CW complex the action G × X → X is cellular: Every cell in X has finite G-stabilizer.
Proof.The action α is Bourbaki-proper if and only if the map α is proper (see Corollary 4) which is equivalent to the statement that for each compact K ⊂ X, the subset (K|K) G × K is compact.Hence, (1) ⇐⇒ (2).Assume that (3) holds, i.e. α is proper, equivalently, the map α is proper.This means that for This subset is closed in G × X and projects onto (K|K) G in the first factor and to the subset in the second factor.Hence, properness of the action α implies finiteness of (K|K) G , i.e. (2).Conversely, if (K|K) G is finite, compactness of g −1 (K) for every g ∈ G implies compactness of the union (⋆).Thus, (2) ⇐⇒ (3).
In order to show that (2)⇒(6), suppose that x, y are G-dynamically related points: There exists a sequence g n → ∞ in G and a sequence x n → x such that g n (x n ) → y.The subset (6)⇒( 5): Suppose that the neighborhoods U x , V y do not exist.Let {U n } n∈N , {V n } n∈N be countable bases at x, y respectively.Then for every n there exists (5)⇒(4).Consider a compact K ⊂ X.Then for each x ∈ K, y ∈ K there exist neighborhoods U x , V y such that (U x |V y ) G is finite.The product sets U x × V y , x, y ∈ K constitute an open cover of K 2 .By compactness of K 2 , there exist x 1 , ..., x n , y 1 , ..., y m ∈ K such that The implication (4)⇒( 2) is immediate.This concludes the proof of equivalence of the properties (1)-( 6).
(5)⇒( 7): Finiteness of G-stabilizers of points in X is clear.Let x, y be points in distinct Gorbits.Let U ′ x , V ′ y be neighborhoods of x, y such that (U ′ x |V ′ y ) G = {g 1 , ..., g n }.For each i, since X is Hausdorff, there are disjoint neighborhoods V i of y and W i of g i (x i ).Now set Then gU x ∩ V y = ∅ for every g ∈ G.
(7)⇒( 6): It is clear that (7) implies that there are no dynamically related points with distinct G-orbits.In particular, every G-orbit in X is closed.
Assume now that X is completely metrizable and G is countable.Suppose that a point x ∈ X is G-dynamically related to itself.Since the stabilizer G x is finite, the point x is an accumulation point of Gx; moreover, Gx is closed in X.Hence, Gx is a closed perfect subset of X.Since X admits a complete metric, so does its closed subset Gx.Thus, for each g ∈ G, the complement U g := Gx − {gx} is open and dense in Gx.By the Baire Category Theorem, the countable intersection g∈G U g is dense in Gx.However, this intersection is empty.A contradiction.
It is clear that (6)⇒( 8) (without any extra assumptions).( 8)⇒ (6).Suppose that X is a metric space and the G-action is equicontinuous.Equicontinuity implies that for each z ∈ X, a sequence z n → z and g n ∈ G, Suppose that there exist a pair of G-dynamically related points x, y ∈ X: ∃x n → x, g n ∈ G, g n x n → y.By the equicontinuity of the action, g n x → y.Since g n → ∞, there exist subsequences g ni → ∞ and g mi → ∞ such that the products h i := g −1 ni g mi are all distinct.Then, by the equicontinuity, The implications (5)⇒( 9)⇒( 8) and ( 5)⇒( 10)⇒( 11) are clear.
Lastly, let us prove the implication (11)⇒(2).We first observe that every CW complex is Hausdorff and 1st countable.Furthermore, every compact K ⊂ X intersects only finitely many open cells e λ in X. (Otherwise, picking one point from each nonempty intersection K ∩ e λ we obtain an infinite closed discrete subset of K.) Thus, there exists a finite subset E := {e λ : λ ∈ Λ} of open cells in X such that for every g ∈ (K|K) G , gE ∩ E = ∅.Now, finiteness of (K|K) G follows from finiteness of cell-stabilizers in G.
Unfortunately, the property that every point of X is a wandering point is frequently taken as the definition of proper discontinuity for G-actions, see e.g.[13,18].Items ( 8) and (10) in the above theorem provide a (weak) justification for this abuse of terminology.I feel that the better name for such actions is wandering actions.
Example 12. Consider the action of G = Z on the punctured affine plane X = R 2 − {(0, 0)}, where the generator of Z acts via (x, y) → (2x, 1 2 y).Then for any p ∈ X, the G-orbit Gp has no accumulation points in X.However, any two points p = (x, 0), q = (0, y) ∈ X are dynamically related.Thus, the action of G is not proper.
This example shows that the quotient space of a wandering action need not be Hausdorff.
Lemma 13.Suppose that G × X → X is a wandering action.Then each G-orbit is closed and discrete in X.In particular, the quotient space X/G is T1.
Proof.Suppose that Gx accumulates at a point y.Then Gx ∩ W y is nonempty, where W y is a G-slice at y.It follows that all points of Gx ∩ W y lie in the same W y -orbit, which implies that Gx ∩ W y = {y}.
There are several reasons to consider proper actions of discrete (and, more generally, locally compact) groups; one reason is that such each proper action of a discrete group yields an orbicovering map in the case of smooth group actions on manifolds: M → M/G is an orbi-covering provided that the action of G on M is smooth (or, at least, locally smoothable).Another reason is that for a proper action on a Hausdorff space, G × X → X, the quotient X/G is again Hausdorff, see Lemma 9.
Question 14. Suppose that G is a discrete group, G × X → X is a free continuous action on an n-dimensional topological manifold X such that the quotient space X/G is a (Hausdorff ) ndimensional topological manifold.Does it follow that the G-action on X is proper?
The answer to this question is negative if one merely assumes that X is a locally compact Hausdorff topological space and X/G is Hausdorff, see [10] (the action given there was even cocompact).Below is a different example.We begin by constructing a non-proper free continuous R-action on a manifold, such that the quotient space is not just Hausdorff but is a manifold with boundary.
Example 15.This is a variation on Example 12.We start with the space Take the quotient space X of Z by the equivalence relation (x, 0) ∼ (0, 1 x ).The space X is homeomorphic to the open Moebius band.The group G = R acts on Z continuously by The above equivalence relation on X is preserved by the G-action and, hence, the G-action descends to a continuous G-action on X.It is easy to see that this action is free but not proper: The equivalence class of (1, 0) is dynamically related to itself.Lastly, the quotient X/G is Hausdorff, homeomorphic to [0, 1) (the equivalence class of (1, 0) maps to 0 ∈ [0, 1)).
Lastly, we use Example 15 to construct a non-proper free Z-action with Hausdorff quotient.We continue with the notation of the previous example.
Example 16.Let Y ⊂ Z denote the following subset of Z (with the subspace topology): Let W denote the projection of Y to X.We take Γ = Z < G = R.This subgroup preserves Y and, hence, W .The quotient W/Γ is homeomorphic to Y ∩ {(0, y) : y ∈ R}, hence, is Hausdorff.At the same time, the Γ-action on W is non-proper.

Cocompactness
There are two common notions of cocompactness for group actions: ( 2), as the image of a compact under the continuous (quotient) map p : X → X/G is compact.
Proof.For each x ∈ X let U x denote a relatively compact neighborhood of x in X.Then Lemma 18. Suppose that X is normal and Hausdorff, G × X → X is a proper action of a discrete group, such that X/G is locally compact.Then X is locally compact.
Proof.Pick x ∈ X.Let W x be a slice for the G-action at x; then W x /G x → X/G is a topological embedding.Thus, our assumptions imply that W x /G x is compact for every x ∈ X.Let (x α ) be a net in W x .Since W x /G x is compact, the net (x α )/G contains a convergent subnet.Thus, after passing to a subnet, there exists g ∈ G x such that (gx α ) converges to some x ∈ W x .Hence, (x α ) subconverges to g −1 (x).Thus, W x is relatively compact.Since X is assumed to be normal, x admits a basis of relatively compact neighborhoods.
Corollary 19.For normal Hausdorff spaces X the two notions of cocompactness agree for proper discrete group actions on X.
On the other hand, if the drop the properness condition, the two notions are not equivalent even for Z-actions with Hausdorff quotients, see the example by R. de la Vega in [27].

Invariant metrics
We start with several general definitions.A discrete subset E of a metric space (X, d) will be called metrically proper if for some (equivalently, every) p ∈ X the function is proper.In other words, every metric ball contains only finitely many points of E. A geodesic metric space, is a metric space (X, d) where every two points x, y are connected by a geodesic segment, i.e. an isometric embedding c : [a, b] → (X, d) such that c(a) = x, c(b) = y.Geodesic segments connecting x to y need not be unique; however, one frequently denotes such segments xy by abusing the notation.We will also conflate geodesic segments and their images.Note that each locally compact complete geodesic metric space (X, d) is proper, i.e. closed metric balls in (X, d) are compact, see [8,Theorem 2.5.28].
An isometric action G × X → X of a discrete group is metrically proper if G acts with finite point-stabilizers and one (equivalently, every) G-orbit in X is a metrically proper subset.In other words, for every x ∈ X the function g → d(x, gx) is proper on G.This condition is stronger than properness of the action but is equivalent to properness of the G-action in the case of proper metric spaces (X, d).Given an isometric properly discontinuous G-action on X we define the function ).If the G-action is metrically proper, then the infimum in the definition of ρ is realized and if the action is also free then ρ([x]) > 0 for all x ∈ X.By abusing the notation, we will also denote this function ρ(x).
Suppose that (X, d) is a metric space and G is a group acting isometrically and metrically properly on X.One defines the quotient-metric d G on X/Γ by 1b. Suppose that (z n ) is a Cauchy sequence in X/G.Then the diameter D of the subset We then inductively lift each geodesic segment in γ to a geodesic segment in X and obtain a piecewise-geodesic path c : [0, T ) → X of length T .Since (X, d) is complete, the path c extends continuously to T .Projecting c(T ) to X/G we obtain the limit of the subsequence (z ni ).Hence, (z n ) converges as well.
2. Suppose that (X, d) is proper.Consider the closed metric ball B( and consider points y, z ∈ B(x, R).We have to verify that . By the triangle inequality, Lastly, for every r > 0 and x ∈ X, q(B(x, r)) is contained in B([x], r) since the quotient map q : (X, d) → (X/G, d G ) is 1-Lipschitz.In the case r = R as above, the fact that q restricts to an isometry on B(x, R) implies the equality q(B(x, R)) = B([x], R).
It turns out that under some rather mild assumptions, given a proper action G × X → X, there is a G-invariant metric metrizing the topology on X: Theorem 22. Suppose that G is a locally compact Hausdorff group, X is locally compact, metrizable space, G × X → X is a proper action and X/G is paracompact.Then X admits a G-invariant metric metrizing the topology on X See [17,Theorem 3].Koszul also notes that if X is paracompact and locally connected, then X/G is paracompact.This theorem was improved in [1]: Theorem 23.Suppose that G is a locally compact Hausdorff group, X is locally compact, σ-compact metrizable space, and G × X → X is a proper action.Then X admits a G-invariant proper metric metrizing the topology on X.
A Riemannian version of these theorems holds in the context of smooth actions of Lie groups: Theorem 24.Suppose that X is a smooth manifold, G is a Lie group and G × X → X is a smooth proper action.There there exists a G-invariant complete Riemannian metric on X.
See [17,Theorem 2] for the existence of an invariant Riemannian metric and [14] for the existence of an invariant complete Riemannian metric.
We next discuss a construction of G-invariant complete geodesic metrics on more general topological spaces.
Theorem 25.Suppose that X is a 2nd countable, connected and locally connected locally compact Hausdorff topological space.Suppose that G × X → X is a proper action of a discrete countable group such that the fixed-point set of each nontrivial element of G is nowhere dense in X.Then X can be metrized using a G-invariant complete geodesic metric. Proof.
Lemma 26.The quotient space Y = X/G is locally compact, connected, locally connected and metrizable.
Proof.Local compactness and connectedness of Y follows from that of X.The 2nd countability of X implies the 2nd countability of Y By Lemma 9, Y is Hausdorff.Since Y is locally compact and Hausdorff, its one-point compactification is compact and Hausdorff, hence, regular.It follows that Y itself is regular.In view of the 2nd countability of Y , Urysohn's metrization theorem implies that Y is metrizable.
Remark 27.Note that each locally compact metrizable space is also locally path-connected.
It is proven in [26] that each locally compact, connected, locally connected metrizable space, such as Y , admits a complete geodesic metric d Y which we fix from now on.Consider the projection p : X → Y .According to [7, Theorem 6.2] (see also [2,Lemma 2]), the map p satisfies the pathlifting property: Given any path c : [0, 1] → Y , a point x ∈ X satisfying p(x) = c(0), there exists a path c : [0, 1] → X such that p • c = c.(This result is, of course, much easier if the G-action is free, i.e. p : X → Y is a covering map.)We let L X denote the set of paths in X which are lifts of rectifiable paths c : [0, 1] → Y .Clearly, the postcomposition of c ∈ L X with an element of G is again in L X .Our next goal is to equip X with a G-invariant length structure using the family of paths L X .Such a structure is a function on L X with values in [0, ∞), satisfying certain axioms that can be found in [8, Section 2.1].Verification of most of these axioms is straightforward, I will check only some (items 1, 2, 3 and 4 below).
1.If c ∈ L X is a lift of a path c in Y , then we declare ℓ(c) to be equal to the length of c. 2. If ci , i = 1, 2, are paths in L X (which are lifts of the paths c 1 , c 2 respectively) whose concatenation b = c1 ⋆ c2 is defined, then b is a lift of the concatenation c 1 ⋆ c 2 .Clearly, ℓ(b) = ℓ(c 1 ) + ℓ(c 2 ).
3. Let U be a neighborhood of some x ∈ X.We need to prove that where the infimum is taken over all γ = c ∈ L X connecting x to points of X \ U .It suffices to prove this claim in the case when and γ connects x to points of ∂U .Then V = p(U ) is a neighborhood of y = p(x) in Y and the paths c = p • γ connect y to points in ∂V .But the lengths of the paths c are clearly bounded away from zero and are equal to the lengths of their lifts c.Thus, we obtain the required bound (28).4. Let us verify that any two points in X are connected by a path in L X .Since X is connected, it suffices to verify the claim locally.Let U is G x -invariant neighborhood of x satisfying (29), such that V = p(U ) is an open metric ball in Y centered at y = p(x).Take u ∈ U , v := p(u) ∈ V .Let c : [0, T ] → V be a geodesic connecting v to y.Then there exists a lift c : [0, T ] → U of c with c(0) = u.Since x ∈ U is the only point projecting to y, we get c(T ) = x.By taking concatenations of pairs of such radial paths in U , we conclude that any two points in U are connected by a path c ∈ L X .
Given length structure on X, one defines a path-metric (metrizing the topology of X) by where the infimum is taken over all γ ∈ L X connecting x 1 to x 2 .By the construction, the projection Lemma 30.The metric d X is complete.
Proof.Let (x n ) be a Cauchy sequence in (X, d X ).By the construction of the metric d X , there exists a finite length path c : [0, 1) → (X, d X ) and a sequence t n ∈ [0, 1) such that c(t n ) = x n , c(0) = x = x 1 .Since the map p is 1-Lipschitz, the path c = p • c : [0, 1) → (Y, d Y ) also has finite length.Since the metric d Y was complete to begin with, the path c extends to a path c : [0, 1] → Y ; set y ′ := c(1).
Assume for a moment that G acts freely on X.Then we have the uniqueness of lifts of paths from Y to X. Thus, the unique lift c of c starting at the point x satisfies the property that its restriction to [0, 1) equals c.It follows that the sequence (x n ) converges to c(1).Below we generalize this argument to the case of non-free actions.
Let U be a neighborhood of y ′ = c(1) which is the projection to Y of a relatively compact slice neighborhood Ũ of some x ′ ∈ p −1 (y ′ ).Without loss of generality (by removing finitely many initial terms of the sequence (x n )) we can assume that the image of the path c lies entirely in U .Applying the path-lifting property to the path c with the prescribed terminal point x ′ , we obtain a lift of the path c that terminates at x ′ .This lift has to be entirely contained in Ũ and its initial point has to be of the form g(x) for some g ∈ G. Applying g −1 to this lift, we obtain another lift of c, denoted c, which starts at x and terminates at g −1 (x ′ ).
Consider the restriction of c to [0, 1).This restriction is also a lift to the path c| [0,1) and the image of the latter lies entirely in U .Hence, the image of c| [0,1) lies entirely in the relatively compact subset g −1 ( Ũ ) ⊂ X.Thus, the Cauchy sequence (x n ) lies in a relatively compact subset of X, and it follows that this sequence converges in X.
Since (X, d X ) is locally compact and complete, by Theorem 2.5.28 (and Remark 2.5.29) in [8], (X, d X ) is a geodesic metric space.Lastly, we note that, by the construction, the length structure on X and, hence, the metric d X , is G-invariant.This concludes the proof of the theorem.
Question 31.Local compactness and local connectivity were critical for the proof of the theorem.Does the theorem hold without these assumptions?7. Fundamental domains of properly discontinuous group actions 7.1.Fundamental sets.As with many notions going back to the 19th century, there is no consistency in the literature regarding the definition of fundamental sets and domains.The next definition follows [17].Our definition is similar to the definition given by Borel and Ji in [5, Definition III.2.14], except that their local finiteness condition is weaker: It is required only for singletons K. Definition 32.A closed subset F ⊂ X is a fundamental set for a proper action of a discrete G on a topological space X if G • F = X and for every compact K ⊂ X, the transporter set (F |K) G is finite (the local finiteness condition).A closed subset F ⊂ X is a fundamental set in the sense of Koszul if, moreover, there exists an open neighborhood U of F such that for every compact K ⊂ X, the transporter set (U |K) G is finite.
Fundamental sets appear naturally in the reduction theory of arithmetic groups (Siegel sets), see [24] and [5].We note, however, that in the literature there are many alternative notions of fundamental sets, inconsistent with the one given above, see e.g.Beardon's book [3, 9.1]: According to Beardon's definition, a subset F of X is called fundamental for the action of G on X if F intersects every G-orbit in X in exactly one point.We will avoid using this definition since its set-theoretic nature provides us with no useful control of the structure of F .
The local finiteness condition in the definition of a fundamental set has several implications: Lemma 33.Suppose that F ⊂ X is a fundamental set for a proper action of a discrete group G on a 1st countable and Hausdorff space X.Then: 1.For every x ∈ X there exists a neighborhood W of x such that (F |U ) G is finite.2. For every x ∈ X there exist a finite subset E = {g 1 , ..., g k } ⊂ G such that the interior of Proof. 1. Suppose that such W does not exist.Then there exists a sequence of distinct elements g n ∈ G and points x n ∈ X such that lim n→∞ x n = x and x n ∈ g n (F ).It follows that for the compact K = {x n : n ∈ N} ∪ {x} the transporter set (F |K) G is infinite, which is a contradiction.2. By the local finiteness condition, there are only finitely many elements g 1 , ..., g k ∈ G such that x ∈ g i (F ).By Part 1 of the lemma, there exists a neighborhood W of x such that W ∩ gF = ∅ only for g ∈ E = {g 1 , ..., g k }.But then, since GF = X, it follows that For each fundamental set F of a G-action on a topological space X we define its quotient space F/G as the quotient space of the equivalence relation x ∼ y ⇐⇒ Gx = Gy.The following proposition explains why fundamental sets are useful: They allow one to describe quotient spaces of proper actions by discrete groups using less information than is contained in the description of the action.
Proposition 34.Suppose that F is a fundamental set for a proper action by discrete group G on a 1st countable and Hausdorff space X.Then the natural projection map p : F/G → X/G is a homeomorphism.
Proof.The map p is continuous by the definition of the quotient topology.It is also obviously a bijection.It remains to show that p is a closed map.Since F is closed, it suffices to show that the projection q : F → X/G is a closed map.Suppose that (x n ) is a sequence in F such that q(x n ) converges to some y ∈ X/G, y is represented by a point x ∈ F .Then there is a sequence h n ∈ G such that z n = h n (x n ) converges to x.If the sequence (h n ) contains infinitely many distinct elements, we obtain a contradiction with the local finiteness property of F similarly to the proof of Lemma 33.Hence, the set E = {h n : n ∈ N} is finite.Applying inverses of the elements h ∈ E, to the sequence (z n ), we see that the subset {x n : n ∈ N} ⊂ X is relatively compact.Thus, q : F → F/G is a closed map.
There are several existence theorems for fundamental sets.The next proposition, proven in [17, Lemma 2], guarantees existence of fundamental sets under the paracompactness assumption on X/G.Proposition 35.Each proper action G × X → X of a discrete group G on a locally compact Hausdorff space X with paracompact quotient X/G admits a fundamental set in the sense of Koszul.
Another construction of fundamental sets is given by closed Dirichlet domains.Let G × X → X be an isometric proper action of a discrete group G on a metric space (X, d).The closed Dirichlet domain for this action is Note that g Dx = Dgx .We also note that Dx is a closed subset of X since it is the intersection of a family of closed subsets {y ∈ X : d(y, x) ≤ d(y, gx)}, g ∈ G.
Proposition 37. Suppose that G × X → X is a metrically proper isometric action of a discrete group G. Then every closed Dirichlet domain D = Dx is a fundamental set for the G-action.
Proof. 1.Let us prove that g D = X.For each y ∈ X the function g → d(y, gx) is a proper function on G, hence, it attains its minimum at some g ∈ G.Then, clearly, y ∈ Dgx = g Dx .Thus, g D = X.
2. Secondly, we verify local finiteness.Consider a metric ball B = B(x, R) for any R > 0. If Dgx ∩ B = ∅, for every point y in this intersection In view of metric properness of the G-action, the set of such elements g ∈ G is finite.
We will discuss Dirichlet domains (and their generalizations via Voronoi tessellations) again in Section 7.2.
Note that in the definition of a closed Dirichlet domain one does not really need a metric, what is needed is a G-invariant continuous function d : X × X → R + .For the proof of Proposition 37 to go through one needs a metric δ on X such that: (a) The G-action is metrically proper on (X, δ).(b) δ(y, x) ≤ φ(d(y, x)) for some function φ.
An example of the situation when this is useful appears in the context of discrete subgroups Γ of G = SL(n, R) acting on the space X of symmetric positive-definite n × n matrices M with det M = 1 by Then Selberg in [23] used the function d : to define an analogue of Dirichlet domains for the Γ-action on X. (See also [15].)The advantage of such generalized Dirichlet domains is that they are intersections of X with polyhedral cones in the space of all symmetric n × n matrices.
Definition 38.Suppose that G × X → X is a continuous action.A closed subset F ⊂ X is a strict fundamental set for the action if it intersects each G-orbit in X in exactly one point.
Strict fundamental sets do not exist often, but they do exist for some classes simplicial group actions on simplicial complexes (one does not even need to assume properness), e.g. for actions of Coxeter groups on Coxeter complexes and actions of semisimple Lie groups (as well as semisimple algebraic groups over discrete valued fields) on buildings (see e.g.[22]).In the next section we will use a construction of strict fundamental sets for properly discontinuous simplicial group actions on vertex sets of connected graphs described below.Suppose that is a simplicial graph (a 1dimensional simplicial complex), G× Γ → Γ is a simplicial action of a discrete group G. (The action need not be proper.)Note that the edge-stabilizers need not fix the invariant edges.However, if Γ ′ denotes the barycentric subdivision of Γ then the induced action of G on Γ ′ is without inversions, i.e. if an element of G preserves an edge, then it fixes the edge pointwise.
Lemma 39.Suppose that Γ is connected.Then there exists a subtree Φ ⊂ Γ ′ such that the vertex set of Φ is a strict fundamental set for the G-action on the vertex set of Γ ′ .
Proof.The quotient Γ ′ /G has natural structure of a connected simplicial graph.Let q : Γ ′ → Γ ′ /G denote the quotient map.Choose T ⊂ Γ ′ /G, a maximal subtree (this may require the Axiom of Choice if the vertex set of Γ ′ /G is uncountable).We will construct Φ by lifting T (inductively) to Γ ′ .We pick a vertex v ∈ Γ ′ /G and lift it arbitrarily to a vertex ṽ ∈ q −1 (v) ∈ Γ ′ .Then, of course, G{ṽ} ∩ {ṽ} = {ṽ}.We proceed inductively, working with subtrees B n ⊂ T which are closed metric balls of radius n centered at v. Suppose that we defined a subtree Φ n ⊂ Γ ′ such that q(Φ n ) = B n and each G-orbit in Γ ′ intersects Φ n in at most one point.Let e = [u, w] be an edge in B n+1 with u ∈ B n .Then there exists an edge ẽ = [ũ, w] of Γ ′ which projects to e and ũ ∈ B n is a vertex projecting to u.We add the edge ẽ (and the vertex w) to Φ n (note that w cannot belong to Φ n ).We repeat this for all edges of B n+1 which are not in B n , resulting in a subtree Φ n+1 ⊂ Γ ′ .By the construction, each G-orbit in Γ ′ intersects Φ n+1 in at most one point.Lastly, the union Φ = n Φ n is a subtree satisfying the required properties.
Note that, unless Γ ′ /G = T (i.e.Γ ′ /G is a tree), Φ is not a fundamental set of the G-action on Γ ′ since preimages of edges of Γ ′ /G that are not in T are not contained in the G-orbit of Φ. 7.2.Fundamental regions and domains.One frequently encounters a sharper version of fundamental sets, called fundamental domains or fundamental regions.Again, there is no consistency in this definition in the literature.Below is a small sample of existing definitions.Ratcliffe in [21, §6.6] defines fundamental regions for a properly discontinuous isometric G-action on a metric space (X, d) as open subsets R ⊂ X such that X = GR = X and gR ∩ R = ∅ for all g ∈ G \ {1}.Then Ratcliffe defines fundamental domains as connected fundamental regions.Ratcliffe also defines locally finite fundamental domains by imposing the extra assumption of local finiteness just as in Definition 32 given above.Beardon in [3, §9.1, 9.2] also defines fundamental domains as open connected subsets as above, but (working in the context of subsets of hyperbolic spaces) imposes the extra condition that the boundary has Lebesgue measure zero.In contrast, S. Katok in §3.1 of [16], defines fundamental regions F ⊂ X as closures of certain open subsets R ⊂ X where R is a fundamental region as in Ratcliffe's definition.Furthermore, Benedetti and Petronio, [4, §C1], define fundamental domains as Borel subsets F ⊂ X such that GF = X and gF ∩ F ⊂ ∂F for all g ∈ G \ {1}.
Below we will adopt a variation of Ratcliffe's and Katok's terminology of fundamental regions/domains but impose the local finiteness condition from the beginning.Definition 40. 1.A subset U of a topological space is called an open domain (or a regular open subset) if U is the interior of its closure.
2. A subset V of a topological space is called a closed domain (or a regular closed subset) if V is the closure of its interior.
Definition 41.Suppose that G × X → X is a proper action of a discrete group on a topological space X.
1.An open subset R ⊂ X is an open fundamental region for this action if the following hold: (1) G • R = X.
(3) For every compact subset K ⊂ X, the transporter set (R|K) G is finite, i.e. the family {gR} g∈G of subsets in X is locally finite.
(3) For every compact subset K ⊂ X, the transporter set (F |K) G is finite.Remark 42.Suppose that G is countable, X is a complete metric space, G×X → X is a continuous action and fixed point sets in X of nontrivial elements of G are nowhere dense.Then Baire's Theorem implies existence of x ∈ X such that G x = {1}.For instance, if X is a connected topological manifold, G is discrete and acts effectively and properly on X.Then the fixed-point set of each nontrivial element of G has empty interior, see [19].
More generally, given a closed discrete subset E ⊂ X, one defines the Voronoi tessellation V E of X corresponding to E. The open/closed tiles of the tessellation are the subsets V x , Vx , x ∈ E, of X defined as The key issue that we will have to deal with is that, even if E is metrically proper, the closed tile Vx is not necessarily the closure of the open tile V x .Moreover, in general, the bisectors Bis(x, z) = {y ∈ X : d(y, x) = d(y, z)} may have nonempty interior in X.This happens, for instance, in the case of metric graphs.
Example 44.Consider the space X which is the union of two coordinate lines in R 2 , with the induced path-metric d, i.e. the restriction of the ℓ 1 -metric from R 2 .Thus, (X, d) is a complete geodesic metric space.Let G = Z 2 , whose generator g acts on X by restriction of the antipodal map (x, y) → (−x, −y) on R 2 .The group G has unique fixed point in X, namely the origin 0 = (0, 0).For every point p ∈ X \ {0} the closed Dirichlet domain Dp is the union of three coordinate rays, while D p consists of just one open coordinate ray.In particular, δD p = Dp \ D p is a coordinate line and, thus, is not contained in the boundary of D p (which is the singleton {0}).The interior of Dp is Dp \ {0}, hence, int Dp ∩ g(int Dp ) is nonempty and equals a coordinate line minus the origin.In particular, the interior of any closed Dirichlet domain cannot be a fundamental region.Note also that the closure of D p is the closed coordinate ray containing p, which implies that GD p = X (it misses two open coordinate rays).Of course, in this example one can take a suitable open subset of int Dp (the union of two open rays) as a fundamental region.However, it cannot be chosen to be connected.Thus, connectedness of fundamental regions (as required by Ratcliffe's definition of a fundamental domain) is an unreasonable requirement in the setting of general complete geodesic metric spaces.
Below we discuss some basic properties of Voronoi tiles.
Lemma 45.Let φ : X → (0, ∞) be an L-Lipschitz function for some L ≤ 1/2 and let E ⊂ X be such that for every x ∈ X, the open ball B(x, φ(x)) has nonempty intersection with E. Then for every x ∈ X, Vx ⊂ B(x, 2φ(x)).
Proof.Take y ∈ Vx .Then there exists z ∈ E such that d(y, z) < φ(y).Since y ∈ Vx , d(x, y) ≤ d(z, y) < φ(y).By the L-Lipschitz property of φ, we have φ(y) ≤ φ(x) + Ld(x, y), implying is the image of E under the quotient map q : X → X/G.We then have two Voronoi tessellations V E (of X) and V S (of X/G equipped with the metric d G ).

Lemma 46. For every closed and every open Voronoi tile
Proof.The statement is a direct consequence of definitions of Voronoi tiles and the metric d G .
Our next goal is to find a condition on E that ensures injectivity of the restriction of q to each Vx .Recall that in Section 6, given an isometric properly discontinuous action G × X → X, we defined a function ρ : X/G → R + (as well as ρ : X → R + ).
We will construct subsets E ⊂ X satisfying the assumptions of Lemma 47 in Lemma 56.
A subset A of a geodesic metric space (X, d) is starlike with respect to a point a ∈ A if for each x ∈ A every geodesic segment ax is contained in A.
Lemma 48.Suppose that (X, d) is a geodesic space and V E be the Voronoi tessellation corresponding to a metrically proper subset E ⊂ X.Then each tile V x , Vx of V E is starlike with respect to its center.
Proof.The proof is essentially the same as the one in [21,Theorem 6.6.13].Take a point z ∈ Vx and let c : [0, T ] → X be a geodesic connecting x to z.Then for each t ∈ [0, T ] and y ∈ E \ {x}, we get (by the triangle inequality) Hence c(t) ∈ Vx and, therefore, Vx is starlike with respect to x.The same argument works for V x .
The basic examples of Voronoi tessellations are when (X, d) is a Euclidean or a real-hyperbolic space; in these cases Voronoi tiles (and, hence, their intersections) are convex.This need not be the case in general even when one works with, say, complex-hyperbolic spaces (see e.g.[12]).Below we will see that some kind of convexity still holds in the case of Voronoi tessellations of Gromov-hyperbolic spaces.
Recall that a subset Y of a geodesic metric space (X, d) is called λ-quasiconvex if every geodesic segment xy with the end-points in Y is contained in the closed λ-neighborhood of Y , i.e. d(z, Y ) ≤ λ for all z ∈ xy.
Proof. 1.This is a direct consequence of Lemma 48 and the definition of δ-hyperbolicity via slimness of geodesic triangles.
2. Take E = {x, y} and the corresponding Voronoi tessellation of X with just two closed tiles, Vx , Vy .Then Bis(x, y) = Vx ∩ Vy .Suppose that points p, q belong to Bis(x, y).Consider a point z on a geodesic pq in X.By Part 1, there exist points x ′ ∈ Vx , y ′ ∈ Vy within distance δ from z.In particular, d(x ′ , y ′ ) ≤ 2δ.Since the geodesic x ′ y ′ connects Vx , Vy , by continuity of the function d(x, •) − d(y, •), there exists z ′ ∈ x ′ y ′ ∈ Bis(x, y).By the triangle inequality, d(z, z ′ ) ≤ 2δ.
Note that the proof of Lemma 48 also shows that for each z ∈ Vx and t < T , either we get the strict inequality d(x, c(t)) < d(y, c(t)), or c(t) belongs to a geodesic yz.If the former case occurs for all y ∈ E \ {x}, we conclude that we get c(t) ∈ V x .In particular, in that case, Vx is the closure of V x .In order to rule out the second possibility (c(t) belongs to a geodesic yz) one has to impose extra restrictions.For instance, geodesics in Riemannian manifolds and, more generally, manifolds with smooth Finsler metrics and Alexandrov spaces satisfy this property.
Corollary 51.Suppose that (X, d) is a metric space with nonbranching geodesics.Then for each Voronoi tessellation V E of X and every x ∈ E, z ∈ Vx and geodesic c : [0, T ] → xz connecting x to z, one has c(t) ∈ V x , t < T .In particular, Vx is the closure of V x .Moreover, V x is an open domain in X.
Proof.The proof is the same as the one in [21,Theorem 6.6.13].Suppose that c(t) / ∈ V x for certain t < T .Then for all s ∈ [t, T ] we have c(s) ∈ δV x = Vx \ V x .Due to the local finiteness of V E , there exist y ∈ E \ {x} and t ′ , t < t ′ < T , such that c(s) ∈ Vy for all s ∈ [t ′ , T ] ⊂ [t, T ].Therefore, for all s ∈ [t ′ , T ] we get s = d(x, c(s)) = d(y, c(s)), and, thus, d(y, c(s)) + d(c(s), z) = d(y, z).In other words, the concatenation of geodesics yc(t ′ ) ⋆ c(t ′ )z is a geodesic γ in (X, d).For the second segment c(t ′ )z of this geodesic we will take the restriction of c to [t ′ , T ].Since y = x, geodesics c and γ have distinct images; on the other hand, they agree on the open subinterval (t ′ , T ).This contradicts the nonbranching assumption.The proof that V x is an open domain in X is similar and we omit it.
Corollary 52 (See Theorem 6.6.13 in [21]).Suppose that (X, d) is a metric space with nonbranching geodesics, G × X → X is an isometric metrically proper action.Then for every x ∈ X with G x = {1} the open Dirichlet domain D x is a connected open fundamental region for the G-action on X.Moreover, Dx is the closure of D x and the interior of Dx is precisely D x .
Proof.We apply Corollary 51 to the Voronoi tessellation V Gx .Corollary 51 implies that Dx is the closure of D x .Connectedness of D x is clear from the same corollary.The fact that G Dx = X follows from Proposition 37. It remains to prove that the interior of Dx is D x .Take z ∈ Dx ∩ Dy , where y ∈ Gx \ {x}.By applying Corollary 51 to the Voronoi tile Dy we see that z belongs to the closure of D y .Since the latter is disjoint from Dx , we conclude that z / ∈ int Dx .
Question 53.Suppose that M is a connected topological manifold.Does M admit a complete geodesic metric with nonbranching geodesics?
In the rest of the section we will prove existence of connected closed fundamental domains for free properly discontinuous actions on geodesic metric spaces by using Voronoi tessellations more (20) d G ([x], [y]) = min g∈G d(x, Gy) = min g,h∈G d(gx, hy), where [x], [y] ∈ X/G are equivalence classes of points x, y ∈ X under the equivalence relation defined by G. Then d G is a metric on X/G which metrizes the quotient topology on X/G, see [21, Theorem 6.6.2].By the construction, the quotient map q Below we describe the most common construction of open fundamental regions, open Dirichlet domains and their variations.Given an isometric metrically proper action of a discrete group G on a metric space (X, d) and a point x ∈ X, one defines the open Dirichlet domain of the action as D x = {y ∈ X : d(y, x) < d(y, gx) ∀g ∈ G \ G x }.Thus, D x ⊂ Dx (see (36)).It is clear from the construction that for every g ∈ G \ G x , gD x ∩ D x = ∅ and gD x = D x for all g ∈ G x .In order to have a chance to get a fundamental region using open Dirichlet domains one has to assume that G x = {1}.

The point x
is the center of the tiles V x , Vx .Each Vx is closed in X (as the intersection of closed subsets).The open tile V x need not be an open subset of X (the intersection of open subsets need not be open).A sufficient condition is that E ⊂ X is metrically proper, see Section 6.Lemma 43.If E ⊂ X is metrically proper then each open tile V x of V Eis an open subset of X and the collection of tiles Vx , x ∈ E, is locally finite.Proof.1.Take y ∈ V x .In view of metric properness of E, the functiond(y, •) − d(y, x) : E \ {x} → R + attains its positive minimum R at some x ′ ∈ E \ {x}.Then B(y, R/2) ⊂ V x .2.Consider a unit ball B = B(z, 1) ⊂ X. Suppose that Vx ∩ B = ∅ for some x ∈ E.Then, whenever Vy ∩ B = ∅, y ∈ E, d(y, z) ≤ d(x, z) + 1.By the metric properness of E, the number of such points y ∈ E is finite.

Definition 50 .
A geodesic metric space (X, d) has nonbranching geodesics if each geodesic c : I → (X, d) is uniquely determined by its restriction to a nonempty open subset of the interval I.