Invariant Random Subgroups Research Articles

Local permutation stability

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. Our notion is weaker than the “permutation stability” introduced by Glebsky–Rivera and Arzhantseva–Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for a finitely generated amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky, and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng’s classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra L(Γ) for countable groups Γ from a dynamical perspective. It is shown that L(Γ) admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of subalgebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant subalgebra.

Approximate homomorphisms and sofic approximations of orbit equivalence relations

Abstract We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

Realizing invariant random subgroups as stabilizer distributions

Suppose that \nu is an ergodic invariant random subgroup of a countable group G such that [N_{G}(H):H ] = n < \infty for \nu -a.e. H \in \operatorname{Sub} . In this paper, we consider the question of whether \nu can be realized as the stabilizer distribution of an ergodic action G \curvearrowright ( X, \mu ) on a standard Borel probability space such that the stabilizer map x \mapsto G_{x} is n -to-one.

Continuum of allosteric actions for non-amenable surface groups

AbstractLet $\Sigma $ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of probability measure preserving ergodic minimal profinite actions for the fundamental group of $\Sigma $ that are topologically free but not essentially free, a property that we call allostery. Moreover, the invariant random subgroups we obtain are pairwise distincts.

Co-spectral radius of intersections

AbstractWe study the behavior of the co-spectral radius of a subgroupHof a discrete group$\Gamma $under taking intersections. Our main result is that the co-spectral radius of an invariant random subgroup does not drop upon intersecting with a deterministic co-amenable subgroup. As an application, we find that the intersection of independent co-amenable invariant random subgroups is co-amenable.

Faithful invariant random subgroups in acylindrically hyperbolic groups

AbstractBuilding on work from Sun and Kechris‐Quorning, we prove that every acylindrically hyperbolic group admits a weakly mixing probability measure preserving action which is faithful but not essentially free. In other words, admits a weakly mixing nontrivial faithful IRS. We also prove that every nonelementary hyperbolic group admits a characteristic random subgroup with the same properties.

Unimodular measures on the space of all Riemannian manifolds

We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain geometric constraints (e.g. bounded geometry) unimodular measures can be used to compactify sets of finite volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits. We develop a structure theory for unimodular measures on $\mathcal M^d$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated `desingularization' of $\mathcal M^d$. We also give a geometric proof of a compactness theorem for unimodular measures on the space of pointed manifolds with pinched negative curvature, and characterize unimodular measures supported on hyperbolic $3$-manifolds with finitely generated fundamental group.

Uncountably many permutation stable groups

In a 1937 paper B. H. Neumann constructed an uncountable family of 2-generated groups. We prove that all of his groups are permutation stable by analyzing the structure of their invariant random subgroups.

Characters and invariant random subgroups of the finitary symmetric group

We will describe the relationship between the indecomposable characters of Fin(N) and its ergodic invariant random subgroups; and we will interpret each Thoma character χ(β;γ) as an asymptotic limit of a naturally associated sequence of characters induced from linear characters of Young subgroups of finite symmetric groups.

Invariant Schreier decorations of unimodular random networks

We prove that every 2d-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group F d . As a corollary we get that every 2d-regular graphing is the local isomorphic image of a graphing coming from a p.m.p. action of F d .

Invariant random subgroups of groups acting on rooted trees

We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly branch group has continuum many distinct atomless ergodic IRS's. This extends a result of Benli, Grigorchuk and Nagnibeda who exhibit a group of intermediate growth with this property.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

AbstractWe prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Measures and stabilizers of group Cantor actions

We consider a minimal equicontinuous action of a finitely generated group \begin{document}$ G $\end{document} on a Cantor set \begin{document}$ X $\end{document} with invariant probability measure \begin{document}$ \mu $\end{document} , and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup \begin{document}$ H $\end{document} of \begin{document}$ G $\end{document} such that the set of points in \begin{document}$ X $\end{document} whose stabilizers are conjugate to \begin{document}$ H $\end{document} has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.

Character rigidity of simple algebraic groups

We prove the following extension of Tits’ simplicity theorem. Let $$k$$ be an infinite field, G an algebraic group defined and quasi-simple over $$k,$$ and $$G(k)$$ the group of $$k$$ -rational points of G. Let $$G(k)^+$$ be the subgroup of $$G(k)$$ generated by the unipotent radicals of parabolic subgroups of G defined over $$k$$ and denote by $$PG(k)^+$$ the quotient of $$G(k)^+$$ by its center. Then every normalized function of positive type on $$PG(k)^+$$ which is constant on conjugacy classes is a convex combination of $$\mathbf {1}_{PG(k)^+}$$ and $$\delta _e.$$ As corollary, we obtain that, when $$k$$ is countable, the only ergodic IRS’s (invariant random subgroups) of $$PG(k)^+$$ are $$\delta _{PG(k)^+}$$ and $$\delta _{\{e\}}.$$ A further consequence is that, when $$k$$ is a global field and G is $$k$$ -isotropic and has trivial center, every measure preserving ergodic action of $$G(k)$$ on a probability space either factorizes through the abelianization $$G(k)_{\mathrm{ab}}$$ or is essentially free.

Stability and invariant random subgroups

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, $\lim_{k\rightarrow\infty}d_{n_k}( \rho_{n_k}(gh),\rho_{n_k}(g)\rho_{n_k}(h))=0$) is close to an actual homomorphism $\varphi_{n_k} \colon\Gamma\rightarrow\operatorname{Sym}(n_k)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P\u{a}unescu showed the same for abelian groups and raised many questions, especially about P-stability of amenable groups. We develop P-stability in general, and in particular for amenable groups. Our main tool is the theory of invariant random subgroups (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups.

Co-induction and invariant random subgroups

In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of \mathbb F_2 which are all invariant and weakly mixing with respect to the action of Aut (\mathbb F_2) . Moreover, for amenable groups \Gamma\leq \Delta , we obtain that the standard co-induction operation from the space of weak equivalence classes of \Gamma to the space of weak equivalence classes of \Delta is continuous if and only if [\Delta :\Gamma]<\infty or core _\Delta(\Gamma) is trivial. For general groups we obtain that the co-induction operation is not continuous when [\Delta:\Gamma]=\infty . This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].

Critical exponents of invariant random subgroups in negative curvature

Let X be a proper geodesic Gromov hyperbolic metric space and let G be a cocompact group of isometries of X admitting a uniform lattice. Let d be the Hausdorff dimension of the Gromov boundary $${\partial X}$$ . We define the critical exponent $${\delta(\mu)}$$ of any discrete invariant random subgroup $${\mu}$$ of the locally compact group G and show that $${\delta(\mu) > \frac{d}{2}}$$ in general and that $${\delta(\mu) = d}$$ if $${\mu}$$ is of divergence type. Whenever G is a rank-one simple Lie group with Kazhdan’s property (T) it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.

On invariant random subgroups of block-diagonal limits of symmetric groups

We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group $G$ admitting only a finite number of ergodic measures on the path-space $X$ of the associated Bratteli digram, we prove that every non-Dirac ergodic invariant random subgroup of $G$ arises as the stabilizer distribution of the diagonal action on $X^n$ for some $n\geq 1$. As a corollary, we establish that every group character $\chi$ of $G$ has the form $\chi(g) = Prob(g\in K)$, where $K$ is a conjugation-invariant random subgroup of $G$.

Invariant random subgroups over non-Archimedean local fields

Let G be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in G are Benjamini–Schramm convergent to the Bruhat–Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work (Abert et al. in Ann Math 185(3):711–790, 2017) from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a variant of the classical Borel density theorem are also extended.