Some remarks on finitarily approximable groups

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. On a related note, we prove that any non-trivial group which can be approximated by finite groups has a non-trivial quotient that can be approximated by finite special linear groups.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. In Section 4 we also use results of Liebeck and Shalev on bounded generation in finite simple groups.

Finitary Approximations of Groups and Their Applications

“…n ∈ N ≥2 and q is a prime power and (n, q) = (2, 2), (2,3), and recall that Fin is the class of all finite groups. In [43] we prove the following result.…”

“…In [30] Howie presented a group which (by a result of Glebsky [24]) turned out not to be approximated by finite nilpotent groups with arbitrary invariant length function. In Sections 2 and 3, we will survey more general results of this type that have been proved recently in [43].…”

“…In this section, we want to survey some recent results that were proved in joint work with Nikolov and Schneider, see [43]. The main insight that helped us was to combine the relationship between soficity and properties of the profinite topology that was established by Glebsky-Rivera [24] with the deep work on finite groups by Nikolov-Segal [44,53].…”

“…Theorem 2.6 ( [43]). Any non-trivial finitely generated Fin-approximated group has a non-trivial PSL-approximated quotient.…”

On compactifications and product‐free sets

“…Concerning ultraproducts of finite groups, Pillay in [, Theorem 3.1] proved that an ultraproduct of finite simple non‐abelian groups has trivial Bohr compactification. More recently, Nikolov, Schneider and Thom [, Theorem 8] have extended Pillay's result, answering a question of Zilber. Theorem If is an ultraproduct of finite groups, then the identity component of is commutative.…”