arXiv research

The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended u…

We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.

Study critical exponents in normal subgroups of higher rank Lie groups.

Researchers prove Fredholm conditions for differential operators on open manifolds.

We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group h…

Minimal lattice generating set size linked to group co-volume.

Study on spectral properties of Riemannian submersions with special fibers.

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds ap…

Let G be a rank 1 simple Lie group and M be a connected orientable aspherical tame manifold. Assume that each end of M has amenable fundamental group. There are several definitions of volume of representations of the fundamental group of M into G. We give a new definition of volume of representations and furthermore, s…

Groups with specific properties have vanishing $\ell^2$-Betti numbers.

The paper explores non-amenability in infinite-type surfaces and graphs.

Paper shows mapping class groups are not extremely amenable except for specific cases.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.

We generalize Luck's Theorem to show that the L^2-Betti numbers of a residually amenable covering space are the limit of the L^2-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L^2 torsion i…

Coxeter groups admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension.

Classifies 4-manifolds with elementary amenable groups and their boundaries.

Surface groups stable in permutations, first non-amenable example.

Measure-scaling quasi-isometries on graphs have specific scaling groups.

Compact leaves with amenable groups are stable under small perturbations.

This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, R…

Integral foliated simplicial volume is zero for certain amenable covers.

We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving or…

New bounds on group asymptotic dimension derived from finitely amenable actions.

New computations show various properties of bounded cohomology in finitely presented groups.

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free …

The paper extends Johnson's characterization of amenable groups to homomorphisms and acyclicity in bounded cohomology.

We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a finitely generated right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.

Researchers redefine $\ell^\infty$-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of ar…

New framework shows $C^*$-simplicity for groups without certain subalgebras.

New theorem on critical exponents for group actions.

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of a line with the total spaces of flat vector bundles over closed infranilmanifolds.

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-in…

The BNS invariant is applied to Kähler groups in new proofs and results.

We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subcl…

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

Free group automorphisms group rigidity proven.

New group $Q$ has unusual properties like no uniform non-amenability.

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of gro…

Study shows complete affine manifolds have zero simplicial volume.

New proof of surface group theorem for 2D Poincaré duality groups.

Study invariant minimizers in convex functions under amenable groups.

Artin groups of hyperbolic type are boundary amenable and have rigid properties.

New results on relative simplicial volume using bounded acyclicity.

Study irregular behavior of ball averages for non-amenable group actions on foliations.

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every s…

Groups with certain properties have invariant subalgebra rigidity.