Volume 22, Issue 1

Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space ℋ{\mathcal{H}}, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that C*⁢(G){C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in R⁢e⁢p⁢(G,ℋ){Rep(G,\mathcal{H})} under the unitary group U⁢(ℋ){U(\mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep⁢(G,ℋ){\mathrm{Rep}(G,\mathcal{H})}.

We observe that Whitehead’s cut-vertex lemma is an immediate consequence of Stallings folds.

The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S4{S^{4}}, and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groups H⁢(r,n,s){H(r,n,s)} are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups H⁢(r,n,s){H(r,n,s)} that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that H⁢(r,n,s){H(r,n,s)} is a 2-generator knot group if and only if it is a torus knot group.

In this paper, we show that the third subgroup of the Andreadakis–Johnson filtration of the automorphism group of a free group coincides with the third group of the lower central series of the IA-automorphism group.

We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x⋅y∈A{x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.

It is shown that each element of the elementary Chevalley group of rank greater than 2 over a ring of stable rank 1 can be expressed as a product of few commutators.

We study the covering radii of 2-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.

The closed subsets ℱ⁢(X){\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which ℱ⁢(X){\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then ℱ⁢(X){\mathcal{F}(X)} is Hausdorff. If X=G{X=G} is a locally compact group, then the subset 𝒮⁢𝒰⁢ℬ⁢(G){\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in ℱ⁢(G){\mathcal{F}(G)}, and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is ℱ⁢(G){\mathcal{F}(G)} (resp. 𝒮⁢𝒰⁢ℬ⁢(G){\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)}) a totally disconnected space?

Given a group G , we write xG{x^{G}} for the conjugacy class of G containing the element x . A famous result of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G′{G^{\prime}} is finite. Recently we showed that if |xG|≤n{|x^{G}|\leq n} for any commutator x , then |G′′|{|G^{\prime\prime}|} is finite and n -bounded. If |xG′|≤n{|x^{G^{\prime}}|\leq n} for any commutator x , then |γ3⁢(G′)|{|\gamma_{3}(G^{\prime})|} is finite and n -bounded. The present article deals with groups in which the conjugacy classes containing squares are finite with bounded size. The following theorem is proved. Let n be a positive integer, G a group and H the subgroup generated by all squares in G . If |xH|≤n{|x^{H}|\leq n} for any square x∈G{x\in G}, then the order of γ3⁢(H){\gamma_{3}(H)} is finite and n -bounded.

In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes.

Let G be a group and n a positive integer. We define Bn⁢(G){B_{n}(G)} to be the intersection of the normalizers of all the non- n -subnormal subgroups of G . We give a new characterization for nilpotent groups in terms of a series defined via Bn⁢(G){B_{n}(G)}.

We introduce G¯{\overline{G}}-fusions of local pointed groups on a block extension A=b⁢𝒪⁢G{A\kern-1.0pt=\kern-1.0ptb\mathcal{O}G}, where H is a normal subgroup of the finite group G , G¯=G/H{\overline{G}=G/H}, and b is a G -invariant block of 𝒪⁢H{\mathcal{O}H}. We show that certain Clifford extensions associated to these pointed groups are invariant under group graded basic Morita equivalences.