Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold Mc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{c^2}$$\end{document} with double simplex D~1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathcal {D}}}}_1$$\end{document}, seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of Mc2=D~1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{c^2}={\widetilde{{\mathcal {D}}}}_1$$\end{document}, leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to D~1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathcal {D}}}}_1$$\end{document}. Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on Mc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{c^2}$$\end{document} as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.


Introduction
By pairing and logical gluing the side (d − 1)-faces (i.e. facets) of an affine d-polytope P, we can obtain a piecewise linear d-manifolds M, where the fundamental group G of M will be finitely generated by the affine mappings I(g 1 , g −1 1 , g 2 , g −1 2 , . . . ) of paired facets. Thus, the G(I)-equivalence classes of (d − 2)-faces, called edges, will be induced, and for each edge class e a (combinatorial) rotational order ν(e) can be prescribed by so-called Poincaré algorithm [12]. Roughly speaking, we go round any edge e from the edge class e in a transversal 2-plane of the combinatorial tiling T := P, G(I) , where we meet all image edge-domains of the edges in class e, glued together. The repetition order ν(e) will sign, when we turn back into the original identity domain of P.
The collection P, G(I, ν(e)) will define the universal covering space tiling T and the fundamental group G up to a presentation. In [12] we characterized in more details whether the tiling T is realizable in a simply connected homogeneous Riemann space. There occur some difficulties already in dimensions d = 3, see also [13,14].
These will be just illustrated by the Gieseking manifold and its related tiling in vertices, then we obtain noncompact Riemann manifold M =P = H 3 /G with isometry group G acting in H 3 without fixed point. The orbit space will be isometric to the identified starting simplex or double-simplex, respectively, as indicated above. The so-called Dehn-surgery, introduced by Thurston [23], deforms the ideal simplices to have special angles different from π/3, such that a starting polyhedronP will be no more fundamental domain of the induced group G(I) in the original sense. However, H 3 /G can be compact cone manifold, especially an orbifold with rotational axis for singular points.
Of course, this cone manifold can also be presented by a compact polyhedron, then with the same volume as the startingP having ideal vertices. We shall compute these angular parameters and the volume ofP by the classical method of N. I. Lobachevsky (see e.g. in [24]). All computations are carried out by Maple 5.0 on the base of the half-space model of the Bolyai-Lobachevsky hyperbolic space H 3 , where the ideal points at infinity are described by C ∪ {∞} =: C ∞ as complex projective line [5,11,18,23]. The interior point (w, ζ) of H 3 over w ∈ z 1 z 2 ⊂ C ∞ (in a half circle) has the third coordinate ζ > 0 with |z 1 − w||w − z 2 | = ζ 2 (see Fig. 1b).

Gieseking manifold and its surgeries
We start with the ideal simplex of H 3 in the half-space model (Fig. 1), where its ideal vertices at infinity are represented by again, it is conjugated to z 2 . We see that p is a "translation", it is We have obtained the Gieseking manifold with one cusp. Other z, as a complex parameter, makes the stabilizer G ∞ to a conformal group with fixed points This line v∞ will not be covered by the G ∞ -images of the simplexS in H 3 . In the model half-space the translations of G ∞ in (2.5) by (2.2) will be similarities of C ∞ with fixed points v, ∞. E.g. z 1 in (2.2) and z 2 in (2.5) are similarityreflections indicated in Figs. 2, 3, 4 and 5. The simple ratio on 01 is u := 1/(1 + |z − 1|). For z * 2 in (2.5) we can write by (2.2) z * 2 : (u, 1) → (u, 1) .
All data can be computed from (2.10), especially the face angles ofS, equal at the opposite edges (Figs. 1a, 2b) (2.11) However, the computer gives more guarantees. In Tables 1 and 2 we have computed by Maple the volume ofS as well for some values of k. We know [24] that the Lobachevsky function provides the volume of the ideal simplex with the above angles. The formal monodromy group G(z 1 , z 2 , k) above has a unified "presentation"  Then we choose a point Z 1 (e.g. the centre of the inscribed ball) in the simplex S = ∞01z and consider the segments Similarly take Z 2 , as the z −1 Then the corresponding curved (bent) surfaces [q −1 ] and its z −1 will be constructed, transversally to the edges ofS (see Figs. 1b, 2, 3, 4 and also [12]).

Figure 7
Gies.2 series to Tables 3 and 4 by cone manifold for k = 3, l = 1 and multiple tiling of C ∞ Of course, (2.14) equivalent to (2.13) if Observe that our compactification procedure works for more general (k, l) as Figs. 3 and 4 indicates. The cusp of our ideal simplexS as a Klein-bottle can be glued by a "solid Klein-bottle" K. Then the splitting effect also occurs. The cusp ofS will be cut along a Klein-bottle surface to get a boundary. Then we glue to this boundary the boundary of K.

The second variant of our cone manifold series
The requirements in (2.9) provide the second root series   Tables 3  and 4 show these, surprisingly a little bit.
This will be geometrically equivalent to Gies.1 by the half-turn symmetry of ideal simplex: 0 ↔ ∞, 1 ↔ z.

Gies.3-4 tend to the regular ideal simplex manifold
The requirements in (2.9) provide the third root series (i.e. the lower integer part of k/2) . (2.16) and the cone manifold (orbifold) series Gies.3. Our Fig. 10 show the case k = 3 and 9. Table 5 gives computer results. See also Figs. 11 and 12.
The fourth root series will be            We do not give here illustration to Gies.4 series, equivalent to the previous one by half-turn symmetry again. As we look at our Tables 1, 2, 4, 5 and 6, and we can prove for orbifolds, k = 2 provides the minimal volume V = 0.6967 . . . . [9] determined the exact lower bound ≈ 0.0390, the next is ≈ 0.0408 for orientable orbifolds in dimension three. Compare that with the half of the Coxeter orthoscheme (5, 3, 5) ≈ 0.0467 and (3, 5, 3) ≈ 0.01953, two-times less than the optimal one, but this orthoscheme has also reflection, as the authors noticed as well. In higher dimensions the problem is open, in general.

Summary
Now we summarize our results to Gieseking manifold. Remark 2.5. The orientable double cover of Gieseking manifold, known as Thurston manifold (or the complement of the figure-eight-knot) has a "manifold surgery" of volume 0.9813688289 . . . , which is known as second minimal one. But the above construction leads to cone manifold surgeries whose volumes tend to zero at the first series; and tend to the original manifold at the second series, respectively.
We found in [16] the third non-orientable double-ideal-regular-simplex-manifold by computer. This has only cone manifold (and orbifold) surgery phenomena, i.e. with multiple (simple) tilings, respectively, that will be discussed and updated in the next Sect. 3.

Figure 13
The non-orientable ideal double-simplex manifold D 1 with 2 cusps, each with Euclidean plane group 4.pg. In lower right it is described after reciprocity transform u → 1/u,

Double-simplex tilings; manifolds, cone mainfolds and orbifolds to non-orientable manifold D 1 in H 3
As we mentioned in the Abstract, a computer program tells us the possible combinatorial face pairings of the ideal double simplex D up to equivariance. The I-paired double simplex D(I) will be denoted simply by D.
Two face pairings I 1 and I 2 of D are called equivariant if there is a face-to-face incidence preserving combinatorial bijection α of AutD, for which it holds AutD is equivariant to the finite spherical isometry group, with the above Macbeth signature (or * 2, 2, 3 in J Conway's notation), it is of order 12, generated by 3 (plane) reflections. see also [11] for interpretation similar to that of this paper. Here we intended for great accuracy.
The double-simplices D 1 , D 2 , D 3 , D 4 , D 6 , having 2 cusp Klein-bottles will serve the same nonorientable manifold M c 2 , as we shall briefly indicate only by figures, but their pairings are not equivariant (see [16] for details).
In Fig. 13 Table 7 The 2-cusped double-simplex manifold M c 2 = D 1 and its surgery orbifolds, the rough angle data are only for information All are π/3 = 60 •   Table 7 continued   Fig. 18a with fundamental group by (3.19).
We do not get any compact hyperbolic manifold.

(3.21)
This D 1 (p, q) will have 2 closed geodesic lines with singularity points of order p, q, respectively (see Fig. 18b).
A lot of parameter pairs (p,p), (q,q) by (3.8), (3.12) and computer solution of (3.13) for z and w provide us with cone manifolds and multiple tilings with formal minimal group presentation by (3.19-3.20), Fig. 18b: The cases p = ±1 or q = ±1 above, lead to one remaining cusp.