Free products in R. Thompson's group \(V\). (Q2849037)

In the 1960's Richard Thompson introduced three finitely presented discrete groups called \(F\), \(T\) and \(V\). Since then these groups attracted a lot of attention and many equivalent definitions for these three groups have been found. The paper under review discusses properties of the group \(V\), where the authors use the description of \(V\) as a subgroup of the group of self-homeomorphisms of the Cantor set.NEWLINENEWLINE In particular the authors explore the class of subgroups of \(V\). It is well-known and easy to see that the class of subgroups of \(V\) contains isomorphic copies of \(V\) and also contains isomorphic copies of all finite groups. In his dissertation C. Roever proves that this class also contains isomorphic copies of all iterated free products of finite groups. The authors of the paper under review extend this result in the following way: First they define a class \(D_{(V,\mathfrak C)}\) containing among others all finite groups, \(\mathbb Z\) and \(\mathbb Q/\mathbb Z\) and then they show that for all subgroups \(G_1,G_2\leq V\) and all members \(H\in D_{(V,\mathfrak C)}\) the groups \(G_1\wr H=\bigoplus_HG_1\rtimes H\) and \(G_1*G_2\) also embed into \(V\).NEWLINENEWLINE The group \(\mathbb Z^2\) does not belong to \(D_{(V,\mathfrak C)}\) and in fact the authors prove that the group \(\mathbb Z^2*\mathbb Z\) does not embed into \(V\). This is of special interest for the study of the complexity of the word problem: In [\textit{D. F. Holt}, \textit{S. Rees}, \textit{C. E. Roever}, and \textit{R. M. Thomas}, J. Lond. Math. Soc., II. Ser. 71, No. 3, 643-657 (2005; Zbl 1104.20033)] the class \(co\mathcal{CF}\) of groups in which the language of all nontrivial words over a finite generating set forms a context-free language is introduced and studied. There the authors conjecture that the class is not closed under free products and in particular that \(\mathbb Z^2*\mathbb Z\) does not belong to \(co\mathcal{CF}\). In [\textit{J. Lehnert} and \textit{P. Schweitzer}, Bull. Lond. Math. Soc. 39, No. 2, 235-241 (2007; Zbl 1166.20025)] it is shown that \(V\) and hence all subgroups of \(V\) belong to \(co\mathcal{CF}\).