Some remarks on finitarily approximable groups

TL;DR

This paper investigates the limitations of approximating certain groups with finite groups, showing that non-trivial finitely generated perfect groups cannot be approximated by finite solvable groups and characterizing which Lie groups embed into finite groups.

Contribution

It provides new results on the non-approximability of perfect groups and characterizes Lie groups embeddable into finite groups with invariant length functions.

Findings

Non-trivial finitely generated perfect groups are not approximable by finite solvable groups.

Only abelian connected Lie groups can embed into a metric ultraproduct of finite groups with invariant length.

The identity component of a compactification of a pseudofinite group must be abelian.

Abstract

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal.