Related papers: Property (T) in $k$-gonal random groups

Consider a class of probability distributions which is dense in the space of all probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence, for every $d\in\mathbb{N}$. Then, we construct various explicit classes of…

Let $\{Z_k\}_{k\geqslant 1}$ denote a sequence of independent Bernoulli random variables defined by ${\mathbb P}(Z_k=1)=1/k=1-{\mathbb P}(Z_k=0)$ $(k\geqslant 1)$ and put $T_n:=\sum_{1\leqslant k\leqslant n}kZ_k$. It is then known that…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

In this series of papers, we study correspondence between the following: (1) large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which…

We extend Friedman's theorem to show that, for any fixed $r>1$, a random $2r$--regular Schreier graph associated with the action of $r$ uniformly random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ has a…

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…

In this paper the limit probabilities of first-order properties are studied. The random graph $G(n,p)$ {\it obeys Zero-One $k$-Law} if for each first-order property with quantifier depth not greater than $k$ its probability tends to 0 or…

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We…

The main goal of the paper is to present a general model theoretic framework to understand a result of Shalev on probabilistically finite nilpotent groups. We prove that a suitable group where the equation $[x_1,\ldots,x_k]=1$ holds on a…

Let L contain only the equality symbol and let L^+ be an arbitrary finite symmetric relational language containing L . Suppose probabilities are defined on finite L^+ structures with ''edge probability'' n^{- alpha}. By T^alpha, the almost…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of…

Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the…

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative…

We prove that there exists a finitely generated group that satisfies a group law with probability 1 but does not satisfy any group law. More precisely, we construct a finitely generated group G in which the probability that a random element…

This paper aims to introduce a more general definition of quasirandom groups and generalize several well-known results in the literature in this new setting. More precisely, let $G$ be a semi-direct product of groups and $X\subseteq G$, we…

It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically…

We prove that the normalized standard generators of the free orthogonal quantum group $O_N^+$ converge strongly to a free semicircular system as $N \to \infty$. Analogous results are obtained for the free unitary quantum groups, and some…

In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words…

We show that the probability for a finitely generated subgroup of the modular group, of size $n$, to be almost malnormal or non-parabolic, tends to 0 as $n$ tends to infinity -- where the notion of the size of a subgroup is based on a…