Classification of Choquet-Deny Lie groups
We characterize connected Lie groups on which all random walks only have trivial bounded harmonic functions.
math.DS↗
arXiv subjects
Joshua Frisch
Publications and source records attributed to Joshua Frisch.
Non-stability of Liouville measures under convex combinations
For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant bounded harmonic function, equivalently, the Poisson boundary is non-trivial) if and only if at least $k$ of them appear in the convex combination. Particularly, our result implies that the set of Liouville measures is not closed under convex combination, which answers a question of Kaimanovich. We also provide a similar result under the additional assumption of finite entropy for those non-hyper-FC-central countable groups with the property that every symmetric, finitely supported probability measure is Liouville. These groups are the only known non-trivial examples of countable groups that admit Liouville measures with finite entropy. Examples include the lamplighter group over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group of finite permutations on $\mathbb{Z}$.
math.GR↗
Flows with minimal subdynamics
Let $\Gamma$ be a countably infinite discrete group. A $\Gamma$-flow $X$ (i.e., a nonempty compact Hausdorff space equipped with a continuous action of $\Gamma$) is called $S$-minimal for a subset $S \subseteq \Gamma$ if the partial orbit $S \cdot x$ is dense for every point $x \in X$. We show that for any countable family $(S_n)_{n \in \mathbb{N}}$ of infinite subsets of $\Gamma$, there exists a free $\Gamma$-flow $X$ that is $S_n$-minimal for all $n \in \mathbb{N}$; additionally, $X$ can be taken to be a subflow of $2^\Gamma$. This vastly generalizes a result of Frisch, Seward, and Zucker, in which each $S_n$ is required to be a normal subgroup of $\Gamma$. As a corollary, we show that for a given Polish $\Gamma$-flow $X$, there exists a free $\Gamma$-flow $Y$ disjoint from $X$ in the sense of Furstenberg if and only if $X$ has no wandering points. This completes a line of inquiry started by Glasner, Tsankov, Weiss, and Zucker. As another application, we strengthen some of the results of Gao, Jackson, Krohne, and Seward on the structure of Borel complete sections. For example, we show that if $B$ is a Borel complete section in the free part of $2^\Gamma$, then every union of sufficiently many shifts of $B$ contains an orbit (previously, this was only known for open sets $B$). Although our main results are purely dynamical, their proofs rely on recently developed machinery from descriptive set-theoretic combinatorics, namely the asymptotic separation index introduced by Conley, Jackson, Marks, Seward, and Tucker-Drob and its links to the Lov\'{a}sz Local Lemma.
math.DS↗
The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions
We show that the Poisson boundary of random walks of finite entropy on Zariski-dense discrete subgroups of semisimple Lie groups equals the Furstenberg boundary of the corresponding symmetric spaces equipped with the hitting measure, without assuming any moment condition on the random walk.
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Non-Realizability of the Poisson Boundary
We show that for any countable group $ G $ equipped with a probability measure $ \mu $, there exists a randomized stopping time $ \tau $ such that $ (G, \mu _{\tau} )$ admits a strictly larger space of bounded harmonic functions than $ (G,\mu) $, unless this space is trivial for all measures on $ G $. In particular, we exhibit an irreducible probability measure on the free group $F_2$ such that the Poisson boundary is strictly larger than the geometric boundary equipped with the hitting measure, resolving a longstanding open problem. As another consequence, there is never a nontrivial universal topological realization of the Poisson boundary for any countable group.
math.GR↗
Hyper-hyperfiniteness and complexity
We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite then the complexity of hyperfinite countable Borel equivalence relations is as high as possible, namely, $\Sigma^1_2$-complete.
math.LO↗
The Poisson boundary of wreath products
We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $\mu$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,\mu)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment.
math.GR↗
The Poisson boundary of hyperbolic groups without moment conditions
We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.
math.GR↗
Poisson boundary of group extensions
Given a finitely generated group, the well-known Stability Problem asks whether the non-triviality of the Poisson-Furstenberg boundary (which is equivalent to the existence of non-constant bounded harmonic functions) depends on the choice of simple random walk on the group. This question was far from being understood even in the class of linear groups. Given an amenable group, e.g. a solvable group, there is no known characterisation, even a conjectural one, of when it admits a simple random walk with non-trivial boundary. We provide a characterisation of groups with non-trivial boundary for finitely generated linear groups of characteristic $p$. We prove in particular that the Stability Problem has a positive answer in this class of groups. For linear groups of characteristic $0$, we prove a sufficient condition for the triviality of the boundary which does not depend on the choice of a simple random walk. We conjecture that our sufficient condition is also necessary. Our arguments are based on a new comparison criterion for group extensions, on new $\Delta$-restriction entropy estimates and a criterion for boundary non-triviality, and on a new "cautiousness" criterion for triviality of the boundary.
math.GR↗
Minimal subdynamics and minimal flows without characteristic measures
Given a countable group $G$ and a $G$-flow $X$, a measure $\mu\in P(X)$ is called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal $G$-flow, for any group $G$, which does not admit a characteristic measure. We construct for every countable group $G$ such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group $G$ and a collection of infinite subgroups $\{\Delta_i: i\in I\}$, when is there a faithful $G$-flow for which every $\Delta_i$ acts minimally?
math.DS↗
Realizations of countable Borel equivalence relations
We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_\sigma$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.
math.LO↗
Lifts of Borel actions on quotient spaces
Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.
math.LO↗
A dichotomy for Polish modules
Let $R$ be a ring equipped with a proper norm. We show that under suitable conditions on $R$, there is a natural basis under continuous linear injection for the set of Polish $R$-modules which are not countably generated. When $R$ is a division ring, this basis can be taken to be a singleton.
math.LO↗
Quotients by countable subgroups are hyperfinite
We show that for any Polish group $G$ and any countable normal subgroup $Γ\triangleleft G$, the coset equivalence relation $G/Γ$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
math.GR↗
Quasi-Regular Sequences
Let $Σ$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$ with characters from $Σ$ is called $r$-quasi-regular, if for each $σ\inΣ$ the ratio of the longest to shortest interval between consecutive occurrences of $σ$ in $s$ is bounded by $r$. In this paper, we answer a question asked by Kempe, Schulman, and Tamuz, and prove that for any probability distribution $\mathbf{p}$ on a finite alphabet $Σ$, there exists a $2$-quasi-regular infinite sequence with characters from $Σ$ and density of characters equal to $\mathbf{p}$. We also prove that as $\left\lVert\mathbf{p}\right\rVert_\infty$ tends to zero, the infimum of $r$ for which $r$-quasi-regular sequences with density $\mathbf{p}$ exist, tends to one. This result has a corollary in the Pinwheel Problem: as the smallest integer in the vector tends to infinity, the density threshold for Pinwheel schedulability tends to one.
math.CO↗
Characteristic measures of symbolic dynamical systems
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.
math.DS↗
Strong amenability and the infinite conjugacy class property
A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.
math.GR↗
Choquet-Deny groups and the infinite conjugacy class property
A countable discrete group $G$ is called Choquet-Deny if for every non-degenerate probability measure $μ$ on $G$ it holds that all bounded $μ$-harmonic functions are constant. We show that a finitely generated group $G$ is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that $G$ is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when $G$ is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.
math.GR↗