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Groups, periodic planes and hyperbolic buildings. (English) Zbl 1086.51013

The author continues her programme of constructing polyhedra with given links. Using polygonal representations introduced by her [Math. Z. 241, 471–478 (2002; Zbl 1020.51009)] she shows that for \(k\geq 1\) compatible sets \({\mathcal G}_1,\dots,{\mathcal G}_k\) of connected bipartite graphs there exists a family of finite polyhedra with \(2k\)-gonal faces and links at vertices isomorphic to the graphs from \({\mathcal G}_1,\dots,{\mathcal G}_k\).
Introducing the concept of periodic planes two applications of this construction are given. Firstly, the author shows that if the universal cover of a polyhedron \(M_2\) constructed in this fashion in case \(k=2\) and equipped with a (locally) Euclidean metric so that all sides of the 4-gonal faces are geodesics and all angles are \(\pi/4\) contains a flat plane, then it also contains a periodic plane, that is, this plane is stabilized by a \(\mathbb Z\oplus\mathbb Z\) subgroup of the fundamental group of \(M_2\).
Secondly, polyhedra whose faces are \(2k\)-gons and whose links are generalized \(m\)-gons with \(km>m+k\) are considered. Every face of the polyhedron is equipped with a hyperbolic metric such that all sides of the polygons are geodesics and all angles are \(\pi/m\). The universal cover of such a polyhedron then is a hyperbolic building. In this case the author calls a tesselated plane periodic if there exists a surface group of genus \(g\geq 1\) which acts uniformly on it. Using Wicks forms the author proves that a right-angled hyperbolic building whose apartments are hyperbolic planes tesselated by polygons with \(2k\) sides, \(k\geq 3\), contains a periodic plane having the action of a surface group of genus \(2k-4\).

MSC:

51E24 Buildings and the geometry of diagrams
20E42 Groups with a \(BN\)-pair; buildings

Citations:

Zbl 1020.51009
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