Volume 25, Issue 2

We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

A group 𝐺 is said to have property R ∞ R_{\infty} if, for every automorphism φ ∈ Aut ⁢ ( G ) \varphi\in\mathrm{Aut}(G) , the cardinality of the set of 𝜑-twisted conjugacy classes is infinite. Many classes of groups are known to have this property. However, very few examples are known for which R ∞ R_{\infty} is geometric , i.e., if 𝐺 has property R ∞ R_{\infty} , then any group quasi-isometric to 𝐺 also has property R ∞ R_{\infty} . In this paper, we give examples of groups and conditions under which R ∞ R_{\infty} is preserved under commensurability. The main tool is to employ the Bieri–Neumann–Strebel invariant.

Recently, I. J. Leary and A. Minasyan [Commensurating HNN extensions: Nonpositive curvature and biautomaticity, Geom. Topol. 25 (2021), 4, 1819–1860] studied the class of groups G ⁢ ( A , L ) G(A,L) defined as commensurating HNN-extensions of Z n \mathbb{Z}^{n} . This class, containing the class of Baumslag–Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups G ⁢ ( A , L ) G(A,L) up to isomorphism.

We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

Let S ⊂ GL n ⁢ ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

A Beauville surface is defined by the action of a finite group (Beauville group) on the product of two compact Riemann surfaces. In this paper, we consider higher products and the possibility of a similar action by a finite group, which we call a generalised Beauville group ; we prove several results regarding the existence and construction of infinite families of generalised Beauville groups and provide a complete classification of the abelian ones; we list all generalised Beauville groups of orders from 1 to 1023.

We give a structural theorem for pseudofinite groups of finite centraliser dimension. As a corollary, we observe that there is no finitely generated pseudofinite group of finite centraliser dimension.

A skew brace is a triplet ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , where ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are groups such that the brace relation x ∘ ( y ⋅ z ) = ( x ∘ y ) ⋅ x - 1 ⋅ ( x ∘ z ) x\circ(y\cdot z)=(x\circ y)\cdot x^{-1}\cdot(x\circ z) holds for all x , y , z ∈ A x,y,z\in A . In this paper, we study the number of finite skew braces ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , up to isomorphism, such that ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are both isomorphic to T n T^{n} with 𝑇 non-abelian simple and n ∈ N n\in\mathbb{N} . We prove that it is equal to the number of unlabeled directed graphs on n + 1 n+1 vertices, with one distinguished vertex, and whose underlying undirected graph is a tree. In particular, it depends only on 𝑛 and is independent of 𝑇.