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Reflection triangles in Coxeter groups and biautomaticity
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Pierre-Emmanuel Caprace
Published/Copyright:
November 18, 2005
Abstract
A Coxeter system (W, S ) is called affine-free if its Coxeter diagram contains no affine subdiagram of rank ≥ 3. Let (W, S ) be a Coxeter system of finite rank (i.e. |S | is finite). The main result is that W is affine-free if and only if W has finitely many conjugacy classes of reflection triangles. This implies that the action of W on its Coxeter cubing (defined by Niblo and Reeves [G. Niblo and L. Reeves. Coxeter groups act on CAT(0) cube complexes. J. Group Theory 6 (2003), 399–413]) is cocompact if and only if (W, S ) is affine-free. This result was conjectured in loc. cit. As a corollary, we obtain that affine-free Coxeter groups are biautomatic.
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Published Online: 2005-11-18
Published in Print: 2005-07-20
Walter de Gruyter GmbH & Co. KG
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- Tri-extraspecial groups
- On commutators in p-groups
- Capability of nilpotent products of cyclic groups
- Products of characters and derived length. II
- Character degree graphs, blocks and normal subgroups
- Reflection triangles in Coxeter groups and biautomaticity
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Articles in the same Issue
- Tri-extraspecial groups
- On commutators in p-groups
- Capability of nilpotent products of cyclic groups
- Products of characters and derived length. II
- Character degree graphs, blocks and normal subgroups
- Reflection triangles in Coxeter groups and biautomaticity
- On algebraic sets over metabelian groups
- Bounded automorphisms and quasi-isometries of finitely generated groups
- Pattern recognition and minimal words in free groups of rank 2
