Residually finite groups that do not virtually have the unique product property

TL;DR

This work constructs a finitely generated residually finite torsion-free group $G$ that persistently contains Promislow's group $P$, answering a question about virtual diffuseness. The authors build this via doubles along carefully chosen separable subgroups: starting with a multiplicative sequence $\vec n$ and the subgroups $H(n)$, they form double covers and residual-finite doubles $D(\vec n)=G *_{\hat H} \bar G$ in which copies of $P$ appear inside. A key mechanism is embedding Promislow-type pieces $P_i$ into the doubles using Klein bottle subgroups $K_i$, and a factorial-divisibility condition on $\vec n$ guarantees that every finite index subgroup of $D(\vec n)$ contains a subgroup isomorphic to $P$; choosing $n_i=2i!$ yields concrete (uncountably many) examples that are residually finite but not virtually having the UPP. The construction also yields nonlinear, amenable groups with rich diversity in quasi-isometry classes, underscoring that residually finite groups can fail to be virtually diffuse or UPPl in substantial ways.

Abstract

We construct a finitely generated residually finite group $G$ with the property that every finite index subgroup of $G$ contains a subgroup isomorphic to Promislow's group. Hence $G$ does not have a finite index subgroup with the unique product property.

Figures (2)

• Figure 1: $\widehat{\mathcal{H}}$ is a double cover of $\mathcal{H}$
• Figure 2: $\widehat{\mathcal{H}}_k$ is a double cover of $\mathcal{H}_k$

Theorems & Definitions (27)

• Definition 1.1: Persistently contains
• Theorem 1.2
• Conjecture 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof : Sketch