Submission date: 17 March 2017

Abstract:

The concept of a C-approximable group, as introduced by Holt and Rees for a class of finite groups C, is a common generalization of the concepts of 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. Finally, 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 the main result of Turing. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal.

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Full text arXiv 1703.06092: pdf, ps.