A universal construction for groups acting freely on real trees. (Q2911583)

In this book the authors construct a class of groups \(\mathcal{RF}(G)\) starting from a collection of functions from compact intervals \([0,\alpha]\subset\mathbb R\) into a (fixed, discrete) group \(G\). An exciting property of this class is that each of its groups comes with a canonical action on an associated \(\mathbb R\)-tree \(X_G\) and that any group acting freely on an \(\mathbb R\)-tree \(T\) embeds in \(\mathcal{RF}(H)\) (for some choice of \(H\)) such that \(T\) embeds into \(X_H\). A second embedding theorem is shown for actions on \(\Lambda\)-trees.NEWLINENEWLINE Among other things the following topics are covered: the bounded subgroups of \(\mathcal{RF}(G)\) are determined, it is shown that \(\mathcal{RF}(G)\) is not generated by its elliptic elements, a continuous analog of the classical conjugacy theorem for free groups is established and various aspects of functoriality of the construction are studied.NEWLINENEWLINE A first appendix A contains some basic material on \(\Lambda\)-trees making the book self-contained while a long list of open problems is contained in appendix B ``in the hope of stimulating further research in what the authors feel is an exciting new area'' (from the preface).