Homological torsion growth in non-normal chains of graphs of free groups

Abstract

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1 \ge G_2 \ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, $\liminf_{n\rightarrow \infty} \frac{\log \vert \mathrm{Tor}_p(G_n^{\mathrm{ab}})\vert}{[G:G_n]} >0$ (where $\mathrm{Tor}_p(G_n^{\mathrm{ab}}) = \{g \in G_n^{\mathrm{ab}} \;\vert\; g \text{ has order a power of } p\}$).