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

A recently fertile strand of research in Group Theory is developing non-abelian analogues of classical combinatorial results for arithmetic Cayley graphs, describing properties such as growth, expansion, mixing, diameter, etc. We consider…

Let $w\in\mathbf F_k$ be a non-trivial word and denote by $w(G)\subseteq G$ the image of the associated word map $w\colon G^k\to G$. Let $G$ be one of the finite groups ${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm GO}_{2m}^\pm(q),{\rm…

Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{1}{2} are finite. We prove the existence of a uniform quantifier elimination procedure for formulas of minimal rank (probably the superstable part of the…

The effective version of Chebotarev's density theorem under the Generalized Riemann Hypothesis and the Artin conjecture (cf. Iwaniec and Kowalski's Analytic Number Theory, 5.13) involves a numerical invariant of a subset $D$ of a finite…

In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of {\it Group Testing} (GT), via the recently developed Madiman-Tetali inequalities. Group Testing concerns itself with…

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…

These notes expand upon our lectures on {\em profinite rigidity} at the international colloquium on randomness, geometry and dynamics, organised by TIFR Mumbai at IISER Pune in January 2024. We are interested in the extent to which groups…

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for…

We obtain a solution to a bordism version of Gromov's linearity problem over a large family of acyclic groups, for manifolds with arbitrary dimension. Every group embeds into some acyclic group in this family. Thus, the linear bordism…

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…

We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected…

We show that Baumslag-Solitar groups are virtually 2-avoidable, that is, they admit finite index subgroup whose first homology is devoid of $\mathbb{Z}_2$ summand. We also prove virtual 2-avoidability for some other classes of one-relator…

We investigate finite torsors over big opens of spectra of strongly $F$-regular germs that do not extend to torsors over the whole spectrum. Let $(R,\mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an algebraically closed…

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators…

We show that a first-order sentence is almost surely true in a random group of density d<1/2 if and only if it is true in a non-abelian free group.

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container…

We prove that the local eigenvalue statistics for $d=1$ random band matrices with fixed bandwidth and, for example, Gaussian entries, is given by a Poisson point process and we identify the intensity of the process. The proof relies on an…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

In this work, we define a family of probability densities involving the generalized trigonometric functions defined by Dr\'abek and Man\'asevich [1], which we name Generalized Trigonometric Densities. We show their relationship with the…

We study Property (T) in the $\Gamma(n,k,d)$ model of random groups: as $k$ tends to infinity this gives the Gromov density model, introduced in [Gro93]. We provide bounds for Property (T) in the $k$-angular model of random groups, i.e. the…