On a new construction in group theory

Abstract

This paper continues the investigation of the groups $\mathcal{RF}(G)$ introduced and studied in [I.M. Chiswell and T.W. Müller, A class of groups with canonical ℝ-tree action, Springer LNM, to appear]. Two new concepts, that of a test function, and that of a pair of locally incompatible (test) functions are introduced, and their theory is developed. As application, we obtain a number of new quantitative as well as structural results concerning $\mathcal{RF}(G)$ and its quotient $\mathcal{RF}(G)/E(G)$ modulo the subgroup E(G) generated by the elliptic elements. Among other things, the cardinality of $\mathcal{RF}(G)$ is determined, and it is shown that both $\mathcal{RF}(G)$ and $\mathcal{RF}(G)/E(G)$ contain large free subgroups, and that their abelianizations both contain a large ℚ-vector space as direct summand.