ONE FAMILY OF HYPERBOLIC ONE FAMILY OF HYPERBOLIC SPACE GROUPS WITH SPACE GROUPS WITH SIMPLICIAL DOMAINSSIMPLICIAL DOMAINS

Milica StojanovMilica Stojanoviićć

• Hyperbolic space groups are isometry groups, acting discontinuously on the hyperbolic 3-space with a compact fundamental domain. One possibility to describe them is to look for the fundamental domains of these groups. Face pairing identifications of a given polyhedron give us generators and relations for a space group by the Poincaré Theorem.

• The simplest fundamental domains are simplices. In the process of classifying the fundamental simplices, it is determined 64 combinatorially different face pairings of fundamental simplices, furthermore 35 solid transitive non-fundamental simplex identifications. I. K. Zhuk has classified Euclidean and hyperbolic fundamental simplices of finite volume up to congruence. An algorithmic procedure is given by E. Molnár and I. Prok and they have summarized all these results, arranging identified simplices into 32 families. Each of them is characterized by the so-called maximal series of simplex tilings. Besides spherical, Euclidean, hyperbolic realizations there exist also other metric realizations in 3-dimensional simply connected homogeneous Riemannian spaces, moreover, metrically non-realizable topological simplex tilings occur as well.

Introduction

E. Molnár and I. Prok, Classification of solid transitive simplex tilings in simply connected 3-spaces, Part I, Colloquia Math. Soc. János Bolyai 63. Intuitive Geometry, Szeged (Hungary), 1991. North-Holland, (1994), 311-362.

E. Molnár, I. Prok and J. Szirmai, Classification of solid transitive simplex tilings in simply connected 3-spaces, Part II, Periodica Math. Hung. 35 (1-2), (1997), 47-94.

E. Molnár, I. Prok and J. Szirmai, Classification of tile-transitive 3-simplex tilings and their realizations in homogeneous spaces, Non-Euclidean Geometries, János Bolyai Memorial Volume, Editors: A. Prékopa and E. Molnár, Mathematics and Its Applications, Vol. 581, Springer (2006), 321–363.

И. К. Жук, Правильные разбиения пространств постоянной кривизны на конгурентные тетраедры, I-II, Акаgемия наук Белорусской ССР, Институт Математики, Минск 1980, 1983. Preprints.

I. K. Zhuk, Fundamental tetrahedra in Euclidean and Lobachevsky spaces, Soviet Math. Dokl. Vol. 270 (1983), No.3, 540-543.

References 1

Let P be a polyhedron in a space S 3 of constant curvature and G be the group generated by the face identifications, satisfying conditions:

• For each face of P there is another face and identifying isometry g which maps onto and P onto Pg , the neighbour of P along .

• The isometry g-1 maps the face onto and P onto , joining the simplex P along .

• Each edge segment e1 from any equivalence class (defined below) is successively surrounded by polyhedra ,

which fill an angular region of measure 2π/ν, with a natural number ν.

Poincaré algorithm:

Then G is a discontinuously acting group on S 3, P is a fundamental domain for G and the cycle relations of type for every equivalence class of edge segments form a complete set of relations for G, if we also

add the relations to the occasional involutive generators .

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E. Molnár, Eine Klasse von hyperbolischen Raumgruppen, Contributions Alg. Geom., 30, (1990), 79-100 .

M. Stojanović, Some series of hyperbolic space groups, Annales Univ. Sci. Budapest, Sect. Math., 36, (1993), 85--102.

M. Stojanović, Hyperbolic realizations of tilings by Zhuk simplices, Matematički Vesnik, 49, (1997), 59—68.

M. Stojanović, Hyperbolic space groups for two families of fundamental simplices, Novi Sad J. Math., 29, 3. (1999), 337--348, XII Yugoslav Geometric Seminar, Novi Sad, Octobar 8-11, 1998.

M. Stojanović, Fundamental simplices with outer vertices for hyperbolic groups and their group extensions for truncations, FILOMAT 24/1 (2010), 1-19

M. Stojanović, Four series of hypergolic space gropus with simplicial domains, and their supergroups, Krag. J.Math. 35, 2. (2011), 303-315.

M. Stojanović, Supergroups for six series of hyperbolic simplex groups, Periodica Math. Hung., (to appear)

References 2

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