Related papers: Lectures on approximate groups and Hilbert's 5th p…
In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of…
Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…
In this survey, we discuss a series of linearization problems--for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of…
We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group. We define almost convergence using topologically invariant means and exhibit two different kinds of…
A result of Pyber states that every finite group $G$ contains an abelian subgroup whose order is quasi-polynomially large in $\lvert G\rvert$. We prove a similar result for $K$-approximate subgroups of solvable groups under only modest…
A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This…
We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable…
This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with…
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial…
We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free} components, where…
We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively…
The idea of applying isoperimetric functions to group theory is due to M.Gromov. We introduce the concept of a ``bicombing of narrow shape'' which generalizes the usual notion of bicombing. Our bicombing is related to but different from the…
This habilitation memoir (in French, submitted in May 2014) is made up of five chapters, each being an introduction to work of the author between 2006 and 2014. The core of the memoir consists of the first three chapters, pertaining to…
Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is…
This expository article builds on lecture notes from a minicourse entitled "Cremona groups and CAT(0) cube complexes" and given by the author as part of the 2023 Riverside Workshop on Geometric Group Theory. It presents recent constructions…
In this paper we consider the existence of dense embeddings of Limit groups in locally compact groups generalizing earlier work of Breuillard, Gelander, Souto and Storm [GBSS] where surface groups were considered. Our main results are…
We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components. We apply our work to obtain similar…
Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true…
We prove a fixed point theorem for the action of certain local monodromy groups on \'etale covers and use it to deduce lower bounds in essential dimension. In particular, we give more geometric proofs of many (but not all) of the results of…
We review in a pedagogical manner some of the efforts aiming to extend the gauge/gravity correspondence to non-conformal supersymmetric gauge theories in four dimensions. After giving a general overview, we discuss in detail two specific…