arXiv subjects

Konrad Wróbel

Publications and source records attributed to Konrad Wróbel.

6 recordsLinked to original sources

arXiv2026

Orbit equivalence and total weak mixing of free group actions

We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in\mathbf{N}\cup\{\infty\}$ is generated by a free totally weak mixing action of $\mathbf{F}_n$. This answers a question of Miller and Tserunyan. The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\not=e\in\mathbf{F}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set.

math.DS↗

arXiv2026

Cost of one-relator groups

For any infinite one-relator group $Γ=\langle S \mid w^m\rangle$, we prove that $\mathrm{cost}(Γ)=|S|-\frac{1}{m}$. For such groups, this gives $β^{(2)}_1(Γ)=\mathrm{cost}(Γ) - 1$, answering a special case of Gaboriau's question on the relationship between cost and first $\ell^2$-Betti number.

math.GR↗

arXiv2026

Exactness and the topology of the space of invariant random equivalence relations

We characterize exactness of a countable group $Γ$ in terms of invariant random equivalence relations (IREs) on $Γ$. Specifically, we show that $Γ$ is exact if and only if every weak limit of finite IREs is an amenable IRE. In particular, for exact groups this implies amenability of the restricted rerooting relation associated to the ideal Bernoulli Voronoi tessellation, the discrete analog of the ideal Poisson Voronoi tessellation.

math.GR↗

arXiv2025

Infinite volume ends of quotient graphs and homogeneous spaces

We introduce the space of infinite volume ends of a locally compact second countable (lcsc) space that admits a Radon measure. In certain cases, this coincides with the classical space of ends. Consider a discrete subgroup $Γ$ of a unimodular lcsc group $G$ that is not coamenable. Assume that $G$ has property (T) and the associated homogeneous space $G/Γ$ is equipped with the Haar measure. We demonstrate that if $G$ is path connected, then $G/Γ$ has exactly one infinite volume end. In a related vein, if $G$ acts transitively on a locally finite connected graph $X$ with compact open vertex stabilizers and the action of the subgroup $Γ$ is free, we show that $X/Γ$ has exactly one end. We also obtain identical results for certain discrete subgroups $Γ$ of nonamenable product groups $G$. These results can be applied to understand ends of Schreier graphs and infinite volume ends of quotients of symmetric spaces of noncompact type. For instance, for symmetric spaces $X$ of noncompact type without real or complex hyperbolic factors, every infinite-covolume quotient $Γ\backslash X$ has exactly one end of infinite Riemannian volume.

math.GR↗

arXiv2024

Measurable splittings and the measured group theoretic structure of wreath products

Let $Γ$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups). We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wrΓ$ and $C\wrΓ$ are measure equivalent (in fact, orbit equivalent) -- this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr Γ$ and $(B\times\mathbf{Z})\wrΓ$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$. On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $Γ$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities.

math.GR↗

arXiv2021

Cost of inner amenable groupoids

Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1.

math.DS↗