Volume 25, Issue 3

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) . C(\Gamma)\simeq C(\mathrm{Aut}^{*}(\Gamma),\Gamma^{*}). In other words, C ⁢ ( Γ ) C(\Gamma) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) \mathrm{Aut}^{*}(\Gamma) . In particular, in the case where Γ = Ψ n , c \Gamma=\Psi_{n,c} is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\} whenever n ≥ 3 n\geq 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega} is the free profinite group on countable number of generators.

Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.

Let 𝐺 be a permutation group on a set Ω, and recall that a base for 𝐺 is a subset of Ω such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of 𝐺, denoted Σ ⁢ ( G ) \Sigma(G) , with vertex set Ω and two vertices adjacent if and only if they form a base for 𝐺. If 𝐺 is transitive, then Σ ⁢ ( G ) \Sigma(G) is vertex-transitive, and it is natural to consider its valency (which we refer to as the valency of 𝐺). In this paper, we present a general method for computing the valency of any finite transitive group, and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser, and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.

Assume that A ⁢ ( G ) A(G) and B ⁢ ( H ) B(H) are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively. Ilie and Spronk have shown that continuous piecewise affine maps α : Y ⊆ H → G \alpha\colon Y\subseteq H\to G induce completely bounded homomorphisms Φ : A ⁢ ( G ) → B ⁢ ( H ) \Phi\colon A(G)\to B(H) and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.