Related papers: On random presentations with fixed relator length

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality…

We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann…

We consider the cyclically presented groups defined by cyclic presentations with $2m$ generators $x_i$ whose relators are the $2m$ positive length three relators $x_ix_{i+1}x_{i+m-1}$. We show that they are hyperbolic if and only if $m\in…

We explore the structure of the p-adic automorphism group Gamma of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension…

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions…

Two models of random cones in high dimensions are considered, together with their duals. The Donoho-Tanner random cone $D_{n,d}$ can be defined as the positive hull of $n$ independent $d$-dimensional Gaussian random vectors. The Cover-Efron…

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all…

In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random…

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for…

Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$-random element in $GL_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\ldots,g_r\in GL_N(K)$ and evaluating…

We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and…

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…

Let ${X}_{k}=(x_{k1}, \cdots, x_{kp})', k=1,\cdots,n$, be a random sample of size $n$ coming from a $p$-dimensional population. For a fixed integer $m\geq 2$, consider a hypercubic random tensor $\mathbf{{T}}$ of $m$-th order and rank $n$…

For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$ and we…

We describe standard forms for elements of the higher-dimensional Thompson groups $nV$ arising from gridding subdivision processes. These processes lead to standard normal form descriptions for elements in these groups, and sizes of these…

Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating…

For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity…

We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a…

Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…