Incoherent Coxeter groups. (Q2790903)

A Coxeter group \(G\) is given by the presentation \(\langle a_1,\ldots,a_r\mid a_i^2,\;(a_ia_j)^{m_{ij}}:1\leq i<j\leq r\rangle\) where \(m_{ij}\in\{2,3,\ldots,\infty\}\) and where \(m_{ij}=\infty\) means no relator of the form \((a_ia_j)^{mij}\). A group \(G\) is called coherent if every finitely generated subgroup of \(G\) is finitely presented. Otherwise, \(G\) is called incoherent.NEWLINENEWLINE Using probabilistic methods the authors prove the following theorem. For each \(M\) there exists \(R=R(M)\) such that if \(K\) is a Coxeter group with \(3\leq m_{ij}\leq M\) and \(\mathrm{rank\,}r\geq R\), then \(K\) is incoherent.