Let $\Gamma$ be a countable group and $\operatorname{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space of all subgroups of $\Gamma$. We call a subgroup $\Delta \in \operatorname{Sub}(\Gamma)$ a boomerang subgroup if for every $\gamma \in \Gamma$, $\gamma^{n_i} \Delta \gamma^{-n_i} \rightarrow \Delta$ for some subsequence $\{n_i \} \subset \mathbb{N}$. Poincar\'{e} recurrence implies that $\mu$-almost every subgroup of $\Gamma$ is a boomerang, with respect to every invariant random subgroup (IRS) $\mu$. We establish for boomerang subgroups many density related properties, most of which are known to hold almost surely for IRSs. Our main result is that if $\mathbb{G}$ is a simple, connected Chevalley group over $\mathbb{Q}$...
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We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree ...
In this paper we study the action of a countable group $\Gamma$ on the space of orders on the group....
An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed pro...
This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. In...
We compute the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the generalized solvable Baumslag-Sol...
Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical per...
Abstract. An invariant random subgroup (IRS) of a countable discrete group Γ is, by definition, a co...
We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed ...
In the 1st paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion...
We show that a free-by-cyclic group with a polynomially growing monodromy is subgroup separable exac...
We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated ...
We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator ...
The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of $\operatorname{Cl}_K...
Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the in...
An invariant random subgroup H≤G is a random closed subgroup whose law is invariant to conjugation b...
We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree ...
In this paper we study the action of a countable group $\Gamma$ on the space of orders on the group....
An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed pro...