Deformations of nearby subgroups and approximate Jordan constants

TL;DR

This paper investigates how near-morphisms and embeddings in Banach Lie groups relate to actual group structures, extending classical results on finite subgroup approximations and Jordan's theorem.

Contribution

It generalizes Turing's results on approximable Lie groups and strengthens bounds on finite subgroups in Banach Lie groups.

Findings

Near-morphisms are uniformly close to true morphisms in Banach Lie groups.

Existence of neighborhoods where approximate embeddings are close to genuine embeddings.

Finite subgroups close to a compact subgroup have bounded indices of abelian normal subgroups.

Abstract

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove that (1) $S$-valued continuous maps from compact groups to $\mathbb{U}$ sufficiently close to being morphisms are uniformly close to morphisms; and (2) for any Lie subgroup $\mathbb{G}\le \mathbb{U}$ there is an identity neighborhood $U\ni 1\in \mathbb{U}$ so that $\mathbb{G}\cdot U\cap S$-valued morphisms (embeddings) from compact groups into $\mathbb{U}$ are close to morphisms (respectively embeddings) into $\mathbb{G}$. This recovers and generalizes results of Turing's to the effect that (a) Lie groups arbitrarily approximable by finite subgroups have abelian identity component and (b) if a Lie group is approximable in this fashion and has a faithful $d$-dimensional representation then it is also so approximable by finite groups with the same property. Another consequence is a strengthening of a prior result stating that finite subgroups in a Banach Lie group sufficiently close to a given compact subgroup thereof admit a finite upper bound on the smallest indices of their normal abelian subgroups (an approximate version of Jordan's theorem on finite subgroups of linear groups).