The geometry of twisted conjugacy classes in wreath products. (Q2908724)

In [Int. J. Algebra Comput. 16, No. 5, 875-886 (2006; Zbl 1150.20014)] \textit{D. Gonçalves} and \textit{P. Wong} proved the following result: Theorem: The lamplighter group \(\mathbb Z_n\wr\mathbb Z\) has the \(R_\infty\) property if and only if \(2\mid n\) or \(3\mid n\).NEWLINENEWLINE The proof uses most combinatorial group theory. The purpose of the paper under review is besides to reprove the result above, but now using geometric group theory, to extend it for other finite groups besides the cyclic groups \(\mathbb Z_n\). Notoriously this is the case if \(\mathbb Z_n\) is replaced by \(S_n\), the symmetric group.NEWLINENEWLINE The main result of the paper is: Theorem: Let \(G\) be a finite group with a unique Sylow 2-group \(S_2\). If \(Z(S_2)\in\mathfrak U\) then \(G\in\mathfrak L\). Similarly, if \(G\) has a unique Sylow \(3\)-group \(S_3\) with \(Z(S_3)\in\mathfrak U\) then \(G\in\mathfrak L\). Here \(\mathfrak U\) (\(\mathfrak L\)), denotes the family of the finite Abelian groups \(G\) (the finite groups) such that \(G\wr\mathbb Z\) has the \(R_\infty\) property, respectively.NEWLINENEWLINE The proof uses geometric group theory techniques. A key step is the use of the graph of the group for one specific set of generators which is identified with the Cayley graph \(D(m,m)\) called Diestel-Leader graphs. These graphs are defined in the manuscript. Finally the paper contains a good and detailed exposition in the introduction how the \(R_\infty\) property of a group is related with fixed point theory.NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].