Some remarks on finitarily approximable groups

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.