Related papers: Virtually free groups are $p$-Schatten stable

We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a $p$-adic analogue of Ulam stability, where we take $GL_n(\mathbb{Z}_p)$ as approximating groups…

We prove an estimate for the rank of the intersection of free subgroups in virtually free groups, which is analogous to the Hanna Neumann inequality for subgroups in a free group and to the S.V. Ivanov estimate for subgroups in free…

We count the finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $\Gamma(H)$, we consider the number of vertices of $\Gamma(H)$ to…

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of…

We prove that the normalized standard generators of the free orthogonal quantum group $O_N^+$ converge strongly to a free semicircular system as $N \to \infty$. Analogous results are obtained for the free unitary quantum groups, and some…

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent…

We say that a finitely generated group $\Gamma$ is self-simulable if every effectively closed action of $\Gamma$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $\Gamma$-subshift of finite type.…

We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only…

We introduce a notion of "local stability in permutations" for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups (LEF). Our notion is weaker than the…

We introduce the stable module $\infty$-category for groups of type $\Phi$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $\Phi$ which act on a tree, we show that the stable module…

We show that in a Morse local-to-global group where stable subgroups are separable, the product of any stable subgroups is separable. As an application, we show that the product of stable subgroups in virtually special groups is separable.

We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero. This…

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of…

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism $\varphi$, an element $x\in G$ and a subset $K\subseteq G$, we say that the relative $\varphi$-order of $g$ in $K$,…

We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated…

Given a compact cube complex $X$ that splits as a graph of virtually special cube complexes. Suppose that the fundamental groups of edge spaces are cyclonormal in the fundamental groups of adjacent vertex spaces. We show that $\pi_1X$ has…

A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is…

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let $p$ be a prime, and let $G$ be a compact $p$-adic analytic group with associated $\mathbb{Q}_p$-Lie algebra $\mathcal{L}(G)$. We prove that $G$…

We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if…

We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for…