We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset A of a finite simple group G is the image of some word map w : G × G → G if and only if (i) A contains the identity and (ii) A is invariant under Aut(G).

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GLd have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:

Theorem 1. A residually finite group of finite rank is virtually locally soluble.

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of