International Journal of Algebra and Computation

We prove that a group with spectrum {1, 2, 3, 4, 5, 8} is locally finite and is isomorphic to M₁₀, where M₁₀ = A₆ ⋅ 2 is the 3-transitive Mathieu group of degree 10, i.e. the corresponding maximal subgroup of the sporadic simple group M₁₁.

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

We compute the cogrowth series for Baumslag–Solitar groups BS(N, N) = 〈a,b|a^Nb = ba^N〉, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x₀, x₁, …, x_N ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly ) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x₀, x₁, …, x_N〉_G, the subgroup generated by x₀, x₁, …, x_N in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We introduce a new invariant of Γ-limit groups called Γ-discriminating complexity. We further show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial.

In this paper, we study the structure of some Noetherian modules over group rings and deduce some statements regarding the structure of the groups involved. More precisely, we consider a module A over a group ring RG with the following property: A is a Noetherian RH-module for every subgroup H, which is not contained in the centralizer C_G(A). If G is some generalized soluble group and R is a locally finite field or some Dedekind domain, we describe the structure of G/C_G(A).