Volume 28, Issue 1

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

For n ≥ 2 n\geq 2 , let G 1 = A 1 ∗ ⋯ ∗ A n G_{1}=A_{1}\ast\dots\ast A_{n} and G 2 = B 1 ∗ ⋯ ∗ B n G_{2}=B_{1}\ast\dots\ast B_{n} where the A i A_{i} ’s and B i B_{i} ’s are non-elementary relatively hyperbolic groups. Suppose that, for 1 ≤ i ≤ n 1\leq i\leq n , the Bowditch boundary of A i A_{i} is homeomorphic to the Bowditch boundary of B i B_{i} . We show that the Bowditch boundary of G 1 G_{1} is homeomorphic to the Bowditch boundary of G 2 G_{2} . We generalize this result to graphs of relatively hyperbolic groups with finite edge groups. This extends Martin–Świątkowski’s work in the context of relatively hyperbolic groups.

In a discrete group generated by hyperplane reflections in the 𝑛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 𝑛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 𝑛-dimensional hyperbolic space without common boundary points have a unique common perpendicular.

Let 𝑘 be a global field of positive characteristic. Let G = SU ⁢ ( 3 ) \mathcal{G}=\mathrm{SU}(3) be the non-split group scheme defined from an (isotropic) hermitian form in three variables. In this work, we describe, in terms of the Euler–Poincaré characteristic, the relative homology groups of certain arithmetic subgroups 𝐺 of G ⁢ ( k ) \mathcal{G}(k) modulo a representative system 𝔘 of the conjugacy classes of their maximal unipotent subgroups. In other words, we measure how far the homology groups of 𝐺 are from being the coproducts of the corresponding homology groups of the subgroups U ∈ U U\in\mathfrak{U} .

In this paper, we study nilpotent Q ⁢ [ x ] \mathbb{Q}[x] -powered groups that satisfy the following property: for some set of primes 𝜔 in Q ⁢ [ x ] \mathbb{Q}[x] , every ω ′ \omega^{\prime} -isolated Q ⁢ [ x ] \mathbb{Q}[x] -subgroup in some family of its Q ⁢ [ x ] \mathbb{Q}[x] -subgroups is finite 𝜔-type separable.

Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT 𝑓 , the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

Let 𝐺 be a finite group. Recall that an 𝐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable 𝐴-group. Assuming that the commuting graph is connected, we show when the derived length of 𝐺 is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.

A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph Γ ⁢ ( C ) \Gamma(\mathcal{C}) based on the conjugacy class 𝒞 of the group 𝐺, which was introduced in our previous work, and the notion of a strongly embedded subgroup of 𝐺. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.