Conformal dimension of hyperbolic groups that split over elementary subgroups (Q2073776)

The conformal dimension is an asymptotic invariant of hyperbolic groups first introduced by \textit{P. Pansu} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 14, No. 2, 177--212 (1989; Zbl 0722.53028)]. In the paper under review the authors compute the conformal dimension of a hyperbolic group that splits as a graph of groups with elementary edge groups in terms of the conformal dimensions of the resulting vertex groups. The main result is Theorem 1.1: Suppose \(G\) is a hyperbolic group, and we are given a graphs of groups decomposition of \(G\), with vertex groups \(\{G_{i}\}\) and all edge groups are elementary. Then if \(G\) is not virtually free, \[ \mathrm{Confdim}\; \delta_{\infty}G = \max \Big \{ \{1\} \cup \{\mathrm{Confdim}\; \delta_{\infty}G_{i} \} \Big\}. \] This theorem enables the authors to resolve a question of \textit{M. Bonk} and \textit{B. Kleiner} (Question 6.1 of [Geom. Topol. 9, 219--246 (2005; Zbl 1087.20033)]), characterising those hyperbolic groups which have conformal dimension equal to one (under the assumption of no 2-torsion). Corollary 1.2: If \(G\) is a hyperbolic group with no 2-torsion and not virtually free, then \( \mathrm{Confdim}\; \delta_{\infty}G =1\) if and only if \(G\) has a hierarchical decomposition over elementary edge groups so that each vertex group is elementary or virtually Fuchsian.