Topographs and groups (Q1196808)

Let \(G\) be a group acting on a set \(S\), let \(B\) be a generating set for \(G\) and let \(\Gamma(G,B,S)\) denote the oriented graph with vertex set \(S\) and arcs \((s,bs)\) for \(s\in S\) and \(b\in B\cup B^{-1}\). (This is a Cayley graph of \(G\) when \(G=S\).) The addition of an altitude function \(t:S\to\{n\in \mathbb{Z}:n\geq n_ 0\}\) (for given \(n_ 0\in \mathbb{Z})\) makes \(\Gamma(G,B,S)\) into a so-called topograph. A topograph is said to be reducible if certain pairs of neighbors of any vertex \(v\) whose altitude is a local maximum can be joined by a path all of whose intermediate vertices \(u\) satisfy \(t(u)<t(v)\). Given a reducible topograph \(\Gamma(G,B,S)\) and \(v\in S\), the main result is a sufficient condition for \(\langle B:{\mathcal R}\rangle\) to be a presentation of the quotient group \(G/N(G_ v)\), where \(N(G_ v)\) denotes the normal closure of the stabilizer of \(v\) in \(G\). Applications are made to dihedral groups, to finite symmetric groups, and (especially) to automorphism groups of free groups of finite rank.