Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary (Q6654948)

The nerve of a Coxeter system \((W,S)\) is the simplicial complex \(L=L(W,S)\) whose vertex set is identified with \(S\) and whose simplices correspond to those subsets \(T\subset S\) for which the special subgroup \(W_{T}\) is finite. The labelled nerve \(L^{\bullet}\) of \((W,S)\) is the nerve \(L\) in which the edges are equipped with labels in such a way that any edge \(\{s, t\}\) has label equal to the exponent \(m_{st}\) from the standard presentation associated to \((W,S)\). The labelled nerve of a Coxeter system is unseparable if it is connected, has no separating simplex, no separating pair of non-adjacent vertices, and no separating labelled suspension (i.e. a full subcomplex which is the labelled join of a simplex and a doubleton). If \(K\) is a finite simplicial complex puncture-respecting cohomological dimension is defined as \(\mathrm{pcd}(K)= \max \{n \mid H_{n}(K) \not =0 \text{ or } H_{n}(K \setminus \sigma)\not =0 \text{ for some } \sigma \in \mathcal{S}(K)\}\), where \(\mathcal{S}(K)\) is the family of all closed simplices of \(K\).\N\NIf \((W,S)\) is an indecomposable Coxeter system such that \(W\) is infinite word hyperbolic, a classical problem is to characterize \((W,S)\) when the Gromov boundary \(\partial W\) is homeomorphic to a given topological space.\N\NIn the paper under review, the authors provide a complete characterization in the case where \(\partial W\) is the Sierpiński curve or the Menger curve. In particular \(\partial W\) is homeomorphic to the Sierpiński curve if and only if \(L^{\bullet}\) is unseparable, planar, and not a 3-cycle and \(\partial W\) is homeomorphic to the Menger curve if and only if \(L^{\bullet}\) is unseparable, \(\mathrm{pcd}(L^{\bullet})=1\), and \(L^{\bullet}\) is not planar.