We study actions of discrete groups on contractible topological spaces in which either (1) all stabilizers lie in the family of subgroups of prime power order or (2) all stabilizers lie in the family of finite subgroups. We compare the classifying spaces for actions with stabilizers in these two families, the Kropholler hierarchies build on these two families, and group cohomology relative to these two families.
We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, Γ)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Γ$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $…
We study the natural action of $S_n$ on the set of $k$-subsets of the set $\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant base. Here a "base" is a set with trivial pointwise stabilizer, "height" is the maximum size of a subset with the property that its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset, and an "irredundant base" can be thought of as a chain of (pointwise) set-stabilizers for which all containments are proper.
We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible $2$-ended graphs. We draw connections to Willis' scale function and re-prove a result by Trofimov.
Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix any point of $\partial S$. We prove that every invariant random subgroup of $G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $(X,μ)$ either has finite stabilizers…
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 bra…
For any right-angled Artin group $A_Γ$ we construct a finite-dimensional space $\mathcal{O}_Γ$ on which the group $\text{Out}(A_Γ)$ of outer automorphisms of $A_Γ$ acts with finite point stabilizers. We prove that $\mathcal{O}_Γ$ is contractible, so that the quotient is a rational classifying space for $\text{Out}(A_Γ)$. The space $\mathcal{O}_Γ$ blends features of the symmetric space of lattices in $\mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $\mathcal{O}_Γ$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cub…
Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube comple…
Let $G$ be a group acting by homeomorphisms on a Hausdorff compact space $Z$. We constructed a new space $X$ that blows up equivariantly the bounded parabolic points of $Z$. This means, roughly speaking, that $G$ acts by homeomorphisms on $X$ and there exists a continuous equivariant map $π: X \rightarrow Z$ such that for every non bounded parabolic point $z \in Z$, $\#π^{-1}(z) = 1$. We use such construction to characterize topologically some spaces that $G$ acts with the convergence property and to construct new convergence actions of $G$ from old ones. As one of the applications, if $G$ is …
