A new family of sofic one-relator groups

TL;DR

This note constructs infinite families of sofic one-relator groups that are not residually finite nor residually solvable. It proves Theorem A by showing that $G_{a,b^n}(l,k)$ is sofic for all nonzero integers $n,l,k$, using a base case as an HNN extension of a Baumslag–Solitar group and a kernel decomposition for $n\ge 2$ yielding a sofic kernel and sofic-by-amenable structure. Theorem C extends these techniques to $G_{a,b^{-n}ab^n}(l,k)$, where the kernel splits as a free product of $n$ copies of an infinite-relator group and the ambient group is shown to be sofic via local Soficity and amalgamation over amenable subgroups. Overall, the paper furnishes a robust method to generate sofic one-relator groups that escape residual finiteness and solvability, enriching the landscape of sofic examples and highlighting the role of HNN/AMC techniques in establishing soficity.

Abstract

We provide an infinite family of sofic one-relator groups that are not residually solvable nor residually finite. The proof is essentially different from the one in [1], as it does not require just Magnus' decompositions.

Theorems & Definitions (9)

• Theorem A
• Corollary B
• Theorem C
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• proof