Related papers: Arithmeticity of groups $\mathbb Z^n\rtimes\mathbb…

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism B_n^1 -> (ZZ/N ZZ)^n rtimes S_n, where B_n^1 is a braid group of type B. The…

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting…

The problem of describing the group of units $\mathcal{U}(\mathbb{Z} G)$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem.…

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for…

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single…

We prove that the reduced group C*-algebras of infinite countable discrete groups having topologically-free extreme boundaries, or more generally groups that satisfy certain combinatorial property including all acylindrically hyperbolic…

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of M. Sapir.

We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric…

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.

Let Z be an algebraic space of finite type over a field, equipped with an action of the multiplicative group $G_m$. In this situation we define and study a certain algebraic space equipped with an unramified morphism to $A^1\times Z\times…

We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup…

Margulis showed that "most" arithmetic groups are superrigid. Platonov conjectured, conversely, that finitely generated linear groups which are superrigid must be of "arithmetic type." We construct counterexamples to Platonov's Conjecture.

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform…

Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain…

Let $\Gamma$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers a…

In this short note we answer to a question of group theory from arXiv:0910.5080. In that paper the author describes the set of realizable Steinitz classes for so-called $A'$-groups of odd order, obtained iterating some direct and semidirect…

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic…