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.
Domains of discontinuity of Lorentzian affine group actions
We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations.
Free representations of outer automorphism groups of free products via characteristic abelian coverings
Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group
Mixing of 3-term progressions in Quasirandom Groups
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_{...
NEGATIVE IMMERSIONS for ONE-RELATOR GROUPS
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...
Sylow branching coefficients and a conjecture of Malle and Navarro
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 ...
On n-dependent groups and fields II
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...
Markov Chains Through Semigroup Graph Expansions (A Survey)
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.
On the profinite rigidity of surface groups and surface words
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.
A Note on Complex-Hyperbolic Kleinian Groups
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.
