A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Abstract

This paper continues the investigation of the groups $\mathcal{RF}(G)$ first introduced in the forthcoming book of Chiswell and Müller “A Class of Groups Universal for Free ℝ-Tree Actions” and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009). We establish a criterion for a family $\{\mathcal{H}_{\sigma}\}$ of hyperbolic subgroups $\mathcal{H}_{\sigma}\leq\mathcal{RF}(G)$ to generate a hyperbolic subgroup isomorphic to the free product of the $\mathcal{H}_{\sigma}$ (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in $\mathcal{RF}(G)$ for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in $\mathcal{RF}(G)$ generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).