Related papers: A sharp threshold for collapse of the random trian…

The random triangular group \Gamma(n,t) is a group given by a presentation P=<S|R>, where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of \Gamma(n,t) is in some…

Let \Gamma(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. \Gamma(n,p) is free and if p >= C log n/n^2 then a.a.s. \Gamma(n,p) has Kazhdan's…

We show that a random group $\Gamma$ in the triangular binomial model $\Gamma(n, p)$ is a.a.s. not left-orderable for $p\in(cn^{-2}, n^{-3/2-\varepsilon})$, where $c, \varepsilon$ are any constants satisfying $\varepsilon>0$,…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

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…

Given a graph $\Gamma$, its auxiliary \emph{square-graph} $\square(\Gamma)$ is the graph whose vertices are the non-edges of $\Gamma$ and whose edges are the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $\Gamma$. We…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

We consider the random right-angled Coxeter group $W_{\Gamma}$ whose presentation graph $\Gamma\sim \mathcal{G}_{n,p}$ is an Erd{\H o}s--R\'enyi random graph on $n$ vertices with edge probability $p=p(n)$. We establish that $p=1/\sqrt{n}$…

The fundamental group of the $2$-dimensional Linial-Meshulam random simplicial complex $Y_2(n,p)$ was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simple connectivity of $Y_2(n,p)$ is about…

We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then…

Let $Y(n, p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial--Meshulam model, and let $Y \sim Y(n, p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa,…

In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph…

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and…

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…

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…

For $k\mid n$ let $Comb_{n,k}$ denote the tree consisting of an $(n/k)$-vertex path with disjoint $k$-vertex paths beginning at each of its vertices. An old conjecture says that for any $k=k(n)$ the threshold for the random graph $G(n,p)$…

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $\Gamma…

A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in…

Let $\Gamma$ be a crystallographic group of dimension $n,$ i.e. a discrete, cocompact subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ = $O(n)\ltimes\mathbb{R}^n.$ For any $n\geq 2,$ we construct a crystallographic group with a trivial…

We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are `jointly connected'. Bollob\'as, Riordan, Slivken and Smith proved that when the two graphs are independent binomial…