Volume 25, Issue 5

This paper proves that, in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.

For a free group F r F_{r} of finite rank r ≥ 2 r\geq 2 and a non-trivial element w ∈ F r w\in F_{r} , the primitivity rank π ⁢ ( w ) \pi(w) is the smallest rank of a subgroup H ≤ F r H\leq F_{r} such that w ∈ H w\in H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ \pi(w)=\infty ). The set of all subgroups of F r F_{r} of rank π ⁢ ( w ) \pi(w) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) \operatorname{Crit}(w) . These notions were introduced by Puder (2014). We prove that there exists an exponentially generic subset V ⊆ F r V\subseteq F_{r} such that, for every w ∈ V w\in V , we have π ⁢ ( w ) = r \pi(w)=r and Crit ⁡ ( w ) = { F r } \operatorname{Crit}(w)=\{F_{r}\} .

We investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.

Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam’s problem.

We construct a 2-generated pro-2 group with full normal Hausdorff spectrum [ 0 , 1 ] [0,1] , with respect to each of the four standard filtration series: the 2-power series, the lower 2-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-2 group with full Hausdorff spectrum with respect to the lower 2-series.

In this article, we study compact Riemann surfaces of genus 𝑔 with an automorphism of prime order g + 1 g+1 . The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine and realise their full automorphism groups and compute their fields of moduli. We also study some aspects of their Jacobian varieties such as isogeny decompositions and complex multiplication. Finally, we determine the period matrix of the Accola–Maclachlan curve of genus four.

In this paper, we compute powers in the wreath product G ≀ S n G\wr S_{n} for any finite group 𝐺. For r ≥ 2 r\geq 2 a prime, consider ω r : G ≀ S n → G ≀ S n \omega_{r}\colon G\wr S_{n}\to G\wr S_{n} defined by g ↦ g r g\mapsto g^{r} . Let P r ⁢ ( G ≀ S n ) := | ω r ⁢ ( G ≀ S n ) | | G | n ⁢ n ! P_{r}(G\wr S_{n}):=\frac{\lvert\omega_{r}(G\wr S_{n})\rvert}{\lvert G\rvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n G\wr S_{n} is an 𝑟-th power. We prove P r ⁢ ( G ≀ S n + 1 ) = P r ⁢ ( G ≀ S n ) P_{r}(G\wr S_{n+1})=P_{r}(G\wr S_{n}) for all n ≢ - 1 ⁢ ( mod ⁢ r ) n\not\equiv-1\ (\mathrm{mod}\ r) if the order of 𝐺 is coprime to 𝑟. We also give a formula for the number of conjugacy classes that are 𝑟-th powers in G ≀ S n G\wr S_{n} .

Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=D\cup D^{-1} for some integer n ≥ 2 n\geq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ \langle K\rangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 n\geq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.

We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system.