Related papers: Random Groups at Density $d<1/2$: Sharp Length Ine…

We study random quotients of a fixed non-elementary hyperbolic group in the Gromov density model. Let $G=\langle S\;\vert\; T\rangle $ be a finite presentation of a non-elementary hyperbolic group, and let $Ann_{l,\omega }(G)$ be the set of…

The standard $(n, k, d)$ model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length $k$ on an $n$-element generating set. Gromov's density model of random groups considers the…

We provide a full and rigorous proof of a theorem attributed to \.Zuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial…

We introduce a new random group model called the square model: we quotient a free group on $n$ generators by a random set of relations, each of which is a reduced word of length four. We prove, as in the Gromov density model, that for…

Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these…

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an…

We study the distribution of products of conjugacy classes in finite simple groups, obtaining various effective uniformity results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers…

In the density model of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l to infinity. If d is a "density" parameter measuring the rate of exponential growth of…

We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz) with $p$…

We prove that a random group, in Gromov's density model with $d < 1/16$ satisfies with overwhelming probability a universal-existential first-order sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a…

We investigate conformal dimension for the class of infinite hyperbolic groups in the Gromov density model $\mathcal{G}^d_{m,l}$ of random groups with $m \geq 2$ fixed generators, density $0 < d < 1/2$ and relator length $l \to \infty$. Our…

We prove that a random group, in Gromov's density model with $d<1/16$, satisfies a universal sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group.

Motivated by the Asymptotic Equipartition Property and its recently discovered role in the cutoff phenomenon, we initiate the systematic study of varentropy on discrete groups. Our main result is an approximate tensorization inequality…

Any positive word comprised of random sequence of tokens form a finite alphabet can be reduced (without change of length) using an appropriate size Braid group relationships. Surprisingly the Braid relations dramatically reduce the…

We extend the polynomial method of Chen--Garza-Vargas--Tropp--van Handel and Magee--Puder--van Handel for operator-norm bounds in random permutation models to the setting where torsion is present. The main new feature is that asymptotic…

Let $G$ be the special linear group $\mathrm{SL}(2,q)$. We show that if $(a_1,\ldots,a_t)$ and $(b_1,\ldots,b_t)$ are sampled uniformly from large subsets $A$ and $B$ of $G^t$ then their interleaved product $a_1 b_1 a_2 b_2 \cdots a_t b_t$…

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $\epsilon > 0$ such that if G is a residually finite group with…

Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a…

We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the…

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant $1-2d$ depending upon the density parameter $d$. This implies in particular a property generalizing the ordinary $C'$…