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

Abstract

This paper continues the investigation of the groups 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 {H σ } of hyperbolic subgroups H σ ≤ RF (G) to generate a hyperbolic subgroup isomorphic to the free product of the H σ (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 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 RF (G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).