Related papers: First $\ell^2$-Betti numbers and proper proximalit…

Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular Bernoulli shifts have independent and identically…

In this paper, we obtain a new proof result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and…

We show that the first $\ell^2$-Betti number of the duals of the free unitary quantum groups is one, and that all $\ell^2$-Betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras.

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.

Let G be a finitely generated group and (G_i) a descending chain of finite index normal subgroups of G. Given a field K, we consider the sequence b_1(G_i;K)/[G:G_i] of normalized first Betti numbers of G_i with coefficients in K, which we…

The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute some computations and complements to the general theory of $L^2$-Betti…

In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular…

A classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We show two methods, one in the measure preserving…

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group…

We study the computability degree of real numbers arising as $L^2$-Betti numbers or $L^2$-torsion of groups, parametrised over the Turing degree of the word problem.

We construct a family of simple, lacunary hyperbolic groups with property $(T)$ that have rational cohomological dimension~$16$ and whose second $\ell^2$-Betti number can be prescribed to be any positive real. Moreover, we construct…

A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not…

This survey is an extended version of a talk during the Arbeitsgemeinschaft on totally disconnected locally compact groups, held in Oberwolfach in October 2014. We explain the definition of l2-Betti numbers of locally compact groups -- both…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

We find particular relations which we call "Bernoulli-type" in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional…

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x…

We generalize a result of R. Thomas to establish the non-vanishing of the first l2-Betti number for a class of finitely generated groups.

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among…

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is…

We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.