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 $p$, ${\mathrm{liminf}}_{n\to \infty }\frac{\log |{\mathrm{Tor}}_p(G_n^{\mathrm{ab}})|}{[G:G_n]}>0$ (where ${\mathrm{Tor}}_p(G_n^{\mathrm{ab}})=\{g\in G_n^{\mathrm{ab}}|g\text{ has order a power of }p\}$).

16 pages, 4 figures, comments are welcome