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Mathematics > Geometric Topology

arXiv:2206.01282 (math)
[Submitted on 2 Jun 2022 (v1), last revised 13 Jul 2022 (this version, v2)]

Title:Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

Authors:Mikhail Belolipetsky, Michael Kapovich
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Abstract:A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.
Comments: 9 pages; v2: revised version, submitted for publication
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
Cite as: arXiv:2206.01282 [math.GT]
  (or arXiv:2206.01282v2 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.2206.01282

Submission history

From: Mikhail Belolipetsky [view email]
[v1] Thu, 2 Jun 2022 20:13:48 UTC (11 KB)
[v2] Wed, 13 Jul 2022 19:08:10 UTC (12 KB)
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