Simple $p$-adic Lie groups with abelian Lie algebras

TL;DR

This paper constructs the first examples of simple, second countable p-adic Lie groups with abelian Lie algebras, answering key questions in the field.

Contribution

It introduces a novel method using generalized small cancellation techniques to build these unique p-adic Lie groups.

Findings

First examples of such groups for all primes and dimensions

Groups are topologically simple and have abelian Lie algebras

Method extends small cancellation to central extensions of hyperbolic groups

Abstract

For each prime $p$ and each positive integer $d$, we construct the first examples of second countable, topologically simple, $p$-adic Lie groups of dimension $d$ whose Lie algebras are abelian. This answers several questions of Gl\"ockner and Caprace-Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.