arXiv subjects

Gil Goffer

Publications and source records attributed to Gil Goffer.

9 recordsLinked to original sources

arXiv2026

Compact Invariant Random Subgroups

We study ergodic invariant random subgroups that give full measure to the subset of compact subgroups. We show that in real Lie groups, compactly generated $p$-adic Lie groups, locally compact hyperbolic groups and infinitely ended groups they are always contained in a compact normal subgroup. In general $p$-adic Lie groups, we show they are contained in the locally elliptic radical. In totally disconnected locally compact groups, we show they are contained in the intersection of all Levi subgroups of inner automorphisms.

math.GR↗

arXiv2024

Asymptotic Burnside laws

We construct novel examples of finitely generated groups that exhibit seemingly-contradicting probabilistic behaviors with respect to Burnside laws. We construct a finitely generated group that satisfies a Burnside law, namely a law of the form $x^n=1$, with limit probability 1 with respect to uniform measures on balls in its Cayley graph and under every lazy non-degenerate random walk, while containing a free subgroup. We show that the limit probability of satisfying a Burnside law is highly sensitive to the choice of generating set, by providing a group for which this probability is $0$ for one generating set and $1$ for another. Furthermore, we construct groups that satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we present a finitely generated group for which every real number in the interval $[0,1]$ appears as a partial limit of the probability sequence of Burnside law satisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar, Gerasimova, and Kozma. The techniques employed in this work draw upon geometric analysis of relations in groups, information-theoretic coding theory on groups, and combinatorial and probabilistic methods.

math.GR↗

arXiv2024

Frattini subgroups of hyperbolic-like groups

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group $G$ admitting a general type action on a hyperbolic space $S$, we show that the induced action of the Frattini subgroup $\Phi(G)$ on $S$ has bounded orbits. This implies that $\Phi(G)$ is "small" compared to $G$; in particular, $|G:\Phi(G)|=\infty$. In contrast, for any finitely generated non-cyclic group $Q$ with $\Phi(Q)=\{ 1\}$, we construct an infinite lacunary hyperbolic group $L$ such that $L/\Phi(L)\cong Q$; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.

math.GR↗

arXiv2023

A note on the Zariski topology on groups

We show that the semigroup Zariski topology on a group can be strictly coarser than the group Zariski topology on it, answering a question of Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse.

math.GR↗

arXiv2023

Probabilistic Burnside groups

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 chosen uniformly from a finite ball in its Cayley graph, or via any non-degenerate random walk, satisfies the group law x^k=1 for some (fixed) integer k, tends to 1. Yet, G contains a non-abelian free subgroup, and therefore G does not satisfy any group law. In particular, this answers two questions of Amir, Blachar, Gerasimova, and Kozma.

math.GR↗

arXiv2022

On compact uniformly recurrent subgroups

Let a group $\Gamma$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural $\Gamma$-action by homeomorphisms. We show that for every minimal $\Gamma$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of U\v{s}akov on compact subgroups whose normalizer is compact.

math.GR↗

arXiv2020

Invariable generation does not pass to finite index subgroups

Using small cancellation methods, we show that the property invariable generation does not pass to finite index subgroups, answering questions of Wiegold and Kantor-Lubotzky-Shalev. We further show that a finitely generated group that is invariably generated is not necessarily finitely invariably generated, answering a question of Cox. The same results were also obtained independently by Minasyan.

math.GR↗

arXiv2019

Conjugacy and Dynamics in Almost Automorphism Groups of Trees

We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson's $V$ by Belk and Matucci. We also analyze dynamics of tree almost automorphisms.

math.GR↗

arXiv2018

A few remarks on invariable generation in infinite groups

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A topological group $G$ is said to be $\mathcal{TIG}$ if it is topologically invariably generated by some subset $S\subseteq G$. In this paper we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group $\mathrm{SL}_{2}(\mathbb{R})$ and the automorphism group of a regular tree are $\mathcal{TIG}$, and that the groups $PSL_m(K) ,m\geq 2$ are not $\mathcal{IG}$ for certain countable fields of infinite transcendence degree over the prime field.

math.GR↗