English

Reflection trees of graphs as boundaries of Coxeter groups

Group Theory 2021-03-10 v1

Abstract

To any finite graph XX (viewed as a topological space) we assosiate some explicit compact metric space Xr(X){\cal X}^r(X) which we call {\it the reflection tree of graphs XX}. This space is of topological dimension 1\le1 and its connected components are locally connected. We show that if XX is appropriately triangulated (as a simplicial graph Γ\Gamma for which XX is the geometric realization) then the visual boundary (W,S)\partial_\infty(W,S) of the right angled Coxeter system (W,S)(W,S) with the nerve isomorphic to Γ\Gamma is homeomorphic to Xr(X){\cal X}^r(X). For each XX, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space Xr(X){\cal X}^r(X).

Keywords

Cite

@article{arxiv.1905.07602,
  title  = {Reflection trees of graphs as boundaries of Coxeter groups},
  author = {Jacek Świątkowski},
  journal= {arXiv preprint arXiv:1905.07602},
  year   = {2021}
}

Comments

53 pages