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The zero divisor conjecture is sufficient to prove for certain class of finitely presented groups where the relations are given by a pairing of generators. We associate Mealy automata to such pairings, and prove that the zero divisor conjecture holds for groups corresponding to invertible automata with three states. In particular, there cannot be zero divisors of support three corresponding to invertible pairings.

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations., Comment: Updated to emphasize the effective nature of the results

We establish results connecting the uniform Liouville property of group actions on the classes of a countable Borel equivalence relation with amenability of this equivalence relation. We also study extensions of Kesten's theorem to certain classes of topological groups and prove a version of this theorem for amenable SIN groups. Furthermore, we discuss relationship between generalizations of Kesten's theorem and anticoncentration inequalities for the inverted orbits of random walks on the classes of an amenable equivalence relation. This allows us to construct an amenable Polish group that does not satisfy the limit conditions in combinatorial extensions of Kesten's theorem., Comment: Significant changes in exposition, results updated, answered one of the questions from the first version

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a F{\o}lner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson's group $F$ and Monod's group of piecewise projective homeomorphisms of the real line., Comment: 32 pages, no figures; v2: referee report taken into account, 36 pages, final version to appear in Commentarii Mathematici Helvetici

This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of 'relative energy' which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szego parameters., Comment: Published version; https://link.springer.com/article/10.1007/s43034-022-00189-2

In this paper we formulate a conjecture which is a strengthening of an extension theorem of Bakonyi and Timotin for positive definite functions on the free group on two generators. We prove that this conjecture implies Connes' embedding conjecture.

Liouville property of actions of discrete groups can be reformulated in terms of existence co-F$\o$lner sets. Since every action of amenable group is Liouville, the property can be served as an approach for proving non-amenability. The verification of this property is conceptually different than finding a non-amenable action. There are many groups that are defined by strongly transitive actions. In some cases amenability of such groups is an open problem. We define $n$-Liouville property of action to be Liouville property of point-wise action of the group on the sets of cardinality $n$. We reformulate $n$-Liouville property in terms of additive combinatorics and prove it for $n=1, 2$. The case $n\geq 3$ remains open.

There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation, we here study elementary amenable bounded automata groups. We are able to isolate the elementary amenable bounded automata groups in three natural subclasses of bounded automata groups. In particular, we show that iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable., Comment: Typos corrected, exposition improved, and proofs clarified

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated., Comment: 9 pages, v2: Typos corrected

We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group $F$ on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group F on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.

We show that Thompson's group $F$ has a topological action on a compact metric space that is proximal and has no fixed points., Comment: Accepted to Ergodic Theory and Dynamical Systems

This is a paper with two aims. First, we show that the map from $\mathbb{Z}/p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}/p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property., Comment: 27 pages. Revised version. We address results by Glebsky and Kassabov-Kuperberg-Riley that go against the heuristic, allowing for the construction of functions that behave like $x\to m^x$, $m\ne 2$ and satisfy a recurrence property

We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups $G

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}\pi(g)=\sum_{h\in H_2}\pi(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}\lambda(g)-\sum_{h\in H_2}\lambda(h)$ coincides with $C^*_\lambda(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_{\lambda}^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above properties. The only currently available tool for proving simplicity of group $C^*$-algebra is Power's condition. We show that it fails for $C_{\lambda}^*(T)$ and present an apparent weakening of that condition which could potentially be used for various new groups $H$ to show the simplicity of $C_{\lambda}^*(H)$. While we use our weakening in the proof of the first result, we also show that the new condition is still too strong to be used to show the simplicity of $C_{\lambda}^*(T)$. Along the way, we give a new application of the Ping-Pong Lemma to find free groups as subgroups in groups of homeomorphisms of the circle generated by elements with rational rotation number.

We give a modification of I. Klep and M. Schweighofer algebraic reformulation of Connes' embedding problem by considering *-algebra of the countably generated free group. This allows to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators., Comment: This is an old paper, which is already appeared in Israel J Math, 2011

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk's group, Basilica group, the full topological group of Cantor minimal system, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, groups that naturally appear in holomorphic dynamics., Comment: The last condition of the main thm was removed, the proofs are streamlined

We characterize groups with Guoliang Yu's property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation., Comment: Final version, to appear in JFA

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x)<\infty$. We study algebraic and analytic properties of $W(X)$ in relation with the metric space structure of $X$, such as amenability of the action of the lamplighter group $ \bigoplus_{X} \mathbf Z/2\mathbf Z \rtimes W(X)$ on $\bigoplus_{X} \mathbf Z/2\mathbf Z$ and property (T)., Comment: 8 pages. v3: final version, with new presentation; to appear in the Bulletin of the BMS

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph $(\Gamma,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(\Gamma,S)

We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space., Comment: The proof has been streamlined: the commutator of W(Z) is not needed anymore

We consider a family of finitely presented groups, called Universal Left Invertible Element (or ULIE) groups, that are universal for existence of one--sided invertible elements in a group ring K[G], where K is a field or a division ring. We show that for testing Kaplansky's Direct Finiteness Conjecture, it suffices to test it on ULIE groups, and we show that there is an infinite family of non-amenable ULIE groups. We consider the Invertibles Conjecture and we show that it is equivalent to a question about ULIE groups. We also show that for any group G, direct finiteness of K[ G x H ] for all finite groups H implies stable finiteness of K[G]. Thus, truth of the Direct Finiteness Conjecture implies stable finiteness. By calculating all the ULIE groups over the field K=F_2 of two elements, for ranks (3,n), n<=11 and (5,5), we show that the Direct Finiteness Conjecture and the Invertibles Conjecture (which implies the Zero Divisors Conjecture) hold for these ranks over F_2., Comment: 44 pages. Version 2 adds a citation and makes minor changes in exposition. Version 3 adds a co-author and the results of computations. Code and raw data associated with the computations have been uploaded with this arXiv submission in the directory ULIE.computations. Retrieve the source code and look in the .tar file for this. (Version 4 is to correct an error in the attachment of this data.)

We show that every logmodular subalgebra of $M_n(\mathbb{C})$ is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured in \cite{VM}. In particular, this shows that every unital contractive representation of a logmodular subalgebra of $M_n(\mathbb{C})$ is automatically completely contractive., Comment: 7 pages

We investigate certain matrices composed of mixed, second-order moments of unitaries. The unitaries are taken from C*-algebras with moments taken with respect to traces, or, alternatively, from matrix algebras with the usual trace. These sets are of interest in light of a theorem of E. Kirchberg about Connes' embedding problem., Comment: 12 pages. In the revision (to version 2) we add some citations to earlier results on extreme correlation matrices that include some results we use