Nonsofic wreath products of residually finite groups

Abstract

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.