## A note on properly discontinuous actions

We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.

We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.

We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations.

Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom groups, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1, A_2, A_3 \subset G$, we have \[ \left|\Pr_{x,y\sim G}\left[ x \in A_1, xy \in A_2, xy^2 \in A_3\right] - \prod_{...

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $w\in F$ be a non-primitive element. The primitivity rank of $w$, $\pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/\langle\langle w\rangle\rangle$ gen...

Funder: Emmanuel College, Cambridge / We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({\bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a prediction by Malle and Navarro from 2012. Our proof of the ...

Abstract: We continue the study ofn-dependent groups, fields and related structures, largely motivated by the conjecture that everyn-dependent field is dependent. We provide evidence toward this conjecture by showing that every infiniten-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Con...

We review the recent approach to Markov chains using the Karnofksy–Rhodes and McCammond expansions in semigroup theory by the authors and illustrate them by two examples.

Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.

Let Γ be a discrete group of isometries acting on the complex hyperbolic n-space HCn. In this note, we prove that if Γ is convex-cocompact, torsion-free, and the critical exponent δ(Γ) is strictly lesser than 2, then the complex manifold HCn/Γ is Stein. We also discuss several related conjectures.