Related papers: Finitely generated simple sharply 2-transitive gro…

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki…

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal…

We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

In the first part of this note, we introduce Tietze transformations for $L$-presentations. These transformations enable us to generalize Tietze's theorem for finitely presented groups to invariantly finitely $L$-presented groups. Moreover,…

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

In this note we give presentations of all finite subgroups of the mapping class group of a closed surface of genus 2 by the Humphries generators up to conjugacy.

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $\Gamma…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

We prove that every profinite group in a certain class with a rational probabilistic zeta function has only finitely many maximal subgroups.

It is shown that infinite, discrete, Kazhdan property (T) groups never have the {\it finite-dimensional density} (FDD) property. This answers a conjecture of Lubotzky and Shalom affirmatively.

We prove that every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $\Gamma$ such that many properties of $G$ are preserved under this embedding. Among those are the properties of being…

We provide an example of a non-finitely generated group which admits a nonempty strongly aperiodic SFT. Furthermore, we completely characterize the groups with this property in terms of their finitely generated subgroups and the roots of…

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…

This is the written version of the Bourbaki seminar given in January 2013 and published in 2014 (modulo an additional early reference added subsequently). It describes the first construction of infinite, finitely generated amenable simple…

We say that a finitely generated group $\Gamma$ is self-simulable if every effectively closed action of $\Gamma$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $\Gamma$-subshift of finite type.…

We prove that finitely generated (not necessarily finitely presented) $C'(\frac{1}{33})$-groups are bi-exact. This is a new class of bi-exact groups.

We classify all finite groups G such that the product of any two non-inverse conjugacy classes of G is always a conjugacy class of G. We also classify all finite groups G for which the product of any two G-conjugacy classes which are not…

We establish a new spectral criterion for Kazhdan's property $(T)$ which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property $(T)$ for the groups $EL_n(R)$, where…

We explore the topological full group [[G]] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [[G]] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale…