群论

We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic…

Graph neural networks (GNNs) have recently been shown to learn algebraic properties of finite groups from their Cayley graphs [1,2]. In this work, we investigate whether such models generalize to infinite finitely generated groups.…

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$. A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ''contained'' in the…

Let $W$ be an even Coxeter group. We prove that among all Coxeter systems generating $W$ the unique even Coxeter system realizes the minimal exponential growth. Our proof relies on comparing the exponential growth rates in the explicit…

We show that the language of geodesic words representing elements of a stable subgroup $H$ of a group $G$ with finite generating set $A$ is regular, and that there is a sublanguage which bijects $H$. Consequently, the growth function of $H$…

Several refinements of (the normality part of) the celebrated It\^o--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex…

We prove a general theorem giving constraints on maps from certain topological groups to inverse limits of bounded torsion groups. From this we obtain some automatic continuity and ultraproduct results. For example, every homomorphism from…

Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to…

We study a weak form of formality for differential graded algebras, called $A_3$-formality, and show that the differential graded $\mathbb{F}_2$-algebras of continuous cochains of all pro-$2$ Demushkin groups are $A_3$-formal. We prove this…

We prove that, for every bounded-degree graph $\Lambda$ admitting a finitely cobounded coarse quasi-action by a group, there is a finitely generated group which does not coarsely embed into $\Lambda$. More generally, for every countable…

We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every…

A group is called bidihedral if it can be expressed as a product of two dihedral subgroups. In this paper, a complete classification for all bidihedral groups is given.

For all Dyer groups, we find an algorithm to determine when two parabolic subgroups are conjugate. Given two conjugate standard parabolic subgroup, we fully describe the conjugating elements in terms of ribbons, showing that the ribbon…

We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L^{2}$-Betti number, then the map that assigns to each surjective integral character the first $L^2$-Betti number…

In their recent paper, Bergfalk and Smythe prove that the isometry equivalence relation on hyperbolic surfaces with finitely-generated fundamental group is concretely classifiable, and ask whether the same result holds true for…

The isometry group of the bounded Urysohn space, $G = \mathrm{Iso}(\U{1})$ is a central object in the study of Polish groups and topological dynamics. It is known that generic sequences in $G$ generate algebraically free dense subgroups. In…

We generalize to certain families of even Artin groups several classical results on right-angled Artin groups. In particular, we compute the cohomology ring, describe the pro-$p$ completion, and determine the $p$-Zassenhaus restricted Lie…

The aim of this short note is to prove an analogue of the existential part of the Schur--Zassenhaus Theorem for finite skew braces: we show that every Hall ideal of a finite skew brace admits a sub-skew brace complement. As an application…

We construct a simple and useful sufficient condition, based on actions on a lattice of idempotents, for monoids admitting homomorphisms to the monogenic free inverse monoid $\mathrm{FIM}(1)$ to not be of type $\mathrm{FP}_2$. This recovers…

For cellular automata over finite alphabets, bijectivity already implies reversibility. Over infinite alphabets this implication may fail, and the remaining obstruction in the periodic case was recorded by Ceccherini-Silberstein and…