Josh Frisch

Portrait of a mathematician smiling. Foreground: white male, mid-20s, medium-length slightly shaggy brown hair, neck framed in hints of a coat. Background: icelandic peninsula in the winter, tundra covered in tan grass and multiple small frozen waterfalls in the background, very low cloud cover. We hope the accessibility is acceptable.

Bio

JOSH FRISCH

I am an Assistant Professor in the Department of Mathematics at University of California San Diego. Previously I was a postdoc at École Normale Supérieure working under Anna Erschler. I received my PhD from Caltech under Alekos Kechris and Omer Tamuz. My main research interests are in the interactions between group theory and dynamics.

I'm also very interested in the interactions between group theory and combinatorics and probability.

I have fairly broad interests in these areas but some particular topics I have enjoyed thinking about include Entropy Theory, Topological Dynamics, Symbolic Dynamics (in particular of Shifts not of Finite Type, and Shifts on Groups), Random Walks on Groups (in particular Poisson Boundaries), Countable Borel (and Measurable) Equivalence Relations, and Infinite Symmetric Graphs.

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Publications

Year	Links	Publication	Abstract
( Last updated Feb 2025 . . . view also on an arXiv search or Google Scholar. )
2024	pdf arXiv	Hyper-hyperfiniteness and complexity with Forte Shinko, Zoltán Vidnyánszky (preprint) Logic	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, $Σ_{2}^{1}$ -complete.
2023	pdf arXiv	The Poisson boundary of wreath products with Eduardo Silva (preprint) Group Theory Dynamical Systems Probability	We give a complete description of the Poisson boundary of wreath products $A ≀ B = ⨁_{B} A ⋊ B$ of countable groups $A$ and $B$ , for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A ≀ B, μ)$ 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 $B = ℤ^{d}$ , $d \geq 3$ , and measures $μ$ with a finite first moment.
2023	pdf arXiv	Poisson Boundary for Upper-Triangular Groups with Anna Erschler, Mark Rychnovsky (preprint) Group Theory Dynamical Systems Probability	We prove that finite entropy random walks on the torsion-free Baumslag group in dimension $d = 2$ have non-trivial Poisson boundary. This is in contrast with the torsion case where the situation for simple random walks on Baumslag groups is the same as for the lamplighter groups of the same dimension. Our proof uses the realization of the Baumslag group as a linear group. We define and study a class of linear groups associated with multivariable polynomials which we denote $G_{k} (p)$ . We show that the groups $G_{3} (p)$ have non-trivial Poisson boundary for all irreducible finite entropy measures, under a condition on the polynomial $p$ which we call the spaced polynomial property. We show that the Baumslag group has $G_{3} (1 + x - y)$ as a subgroup, and that the polynomial $p = 1 + x - y$ , satisfies this property. Given any upper-triangle group of characteristic zero, we prove that one of the following must hold: 1) all finite second moment symmetric random walks on $G$ have trivial boundary 2) the group admits a block, which has a $3$ dimensional wreath product as a subgroup, and all non-degenerate random walks on $G$ have non-trivial boundary. 3) $G$ has a group $G_{3} (p)$ as a subgroup. We give a conjectural characterisation of all polynomials satisfying the spaced polynomial property. If this is confirmed, our result provides a characterisation of linear groups $G$ which admit a finitely supported symmetric random walk with non-trivial boundary.
2022	pdf arXiv	Poisson boundary of group extensions with Anna Erschler (preprint) Group Theory Dynamical Systems Probability	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 $Δ$ -restriction entropy estimates and a criterion for boundary non-triviality, and on a new "cautiousness" criterion for triviality of the boundary.
2021	pdf arXiv	Realizations of countable Borel equivalence relations with Alexander Kechris, Forte Shinko, Zoltán Vidnyánszky (preprint) Logic Dynamical Systems	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 $K_{σ}$ 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.
2019	pdf arXiv	Quasi-Regular Sequences with Wade Hann-Caruthers, Pooya Vahidi Ferdowsi (preprint) Combinatorics	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 $𝐩$ on a finite alphabet $Σ$ , there exists a $2$ -quasi-regular infinite sequence with characters from $Σ$ and density of characters equal to $𝐩$ . We also prove that as ${\| 𝐩 \|}_{\infty}$ tends to zero, the infimum of $r$ for which $r$ -quasi-regular sequences with density $𝐩$ 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.
2022	pdf arXiv	The Poisson boundary of hyperbolic groups without moment conditions with Kunal Chawla, Behrang Forghani, Giulio Tiozzo Accepted to Annals of Probability. Group Theory Dynamical Systems Geometric Topology Probability	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.
2024	pdf arXiv journal	Minimal subdynamics and minimal flows without characteristic measures with Brandon Seward, Andy Zucker Forum of Mathematics, Sigma. vol.12 (May 2024) pp.e58 Dynamical Systems	Given a countable group $G$ and a $G$ -flow $X$ , a measure $μ \in P (X)$ 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 ${{Δ}_{i} : i \in I}$ , when is there a faithful $G$ -flow for which every $Δ_{i}$ acts minimally?
2023	pdf arXiv	Quotients by countable normal subgroups are hyperfinite with Forte Shinko Groups Geometry and Dynamics. vol.17 (Jul 2023) pp.985-992 Group Theory Dynamical Systems Logic	We show that for any Polish group $G$ and any countable normal subgroup $Γ ◁ G$ , the coset equivalence relation $G / Γ$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
2023	pdf arXiv	A dichotomy for Polish modules with Forte Shinko Israel Journal of Mathematics. vol.254, issue 1 (Apr 2023) pp.97-112 Logic Functional Analysis Group Theory Rings and Algebras	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.
2022	pdf arXiv	Lifts of Borel actions on quotient spaces with Alexander Kechris, Forte Shinko Israel Journal of Mathematics (Special Volume for Benjamin Weiss). vol.251, issue 2 (Dec 2022) pp.379-421 Logic Dynamical Systems	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.
2022	pdf arXiv journal	Characteristic measures of symbolic dynamical systems with Omer Tamuz Ergodic Theory and Dynamical Systems. vol.42, issue 5 (May 2022) pp.1655-1661 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.
2019	pdf arXiv journal	Choquet-Deny groups and the infinite conjugacy class property with Yair Hartman, Omer Tamuz, Pooya Vahidi Ferdowsi Annals of Mathematics. vol.190, issue 1 (Jul 2019) pp.307-320 Group Theory Dynamical Systems Probability	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.
2019	pdf arXiv journal	Strong amenability and the infinite conjugacy class property with Omer Tamuz, Pooya Vahidi Ferdowsi Inventiones Mathematicae. vol.218 (Dec 2019) pp.833-851 Group Theory Dynamical Systems	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.
2019	pdf arXiv journal	Normal amenable subgroups of the automorphism group of the full shift with Tomer Schlank, Omer Tamuz Ergodic Theory and Dynamical Systems. vol.39, issue 5 (May 2019) pp.1290-1298 Dynamical Systems	We show that every normal amenable subgroup of the automorphism group of the full shift is contained in its center. This follows from the analysis of this group's Furstenberg topological boundary, through the construction of a minimal and strongly proximal action. We extend this result to higher dimensional full shifts. This also provides a new proof of Ryan's Theorem and of the fact that these groups contain free groups.
2017	pdf arXiv journal	Symbolic dynamics on amenable groups: the entropy of generic shifts with Omer Tamuz Ergodic Theory and Dynamical Systems. vol.37, issue 4 (Jun 2017) pp.1187-1210 Dynamical Systems Group Theory	Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$ . We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$ . The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that for every entropy value $c \in 0, \log \| A \|$ there is a weakly mixing subshift of $A^{G}$ with entropy $c$ . We also show that the set of strongly irreducible shifts does not form a $G_{δ}$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.
2016	pdf arXiv journal	Transitive graphs uniquely determined by their local structure with Omer Tamuz Proceedings of the American Mathematical Society. vol.144 (Nov 2016) pp.1913-1918 Group Theory Metric Geometry	We show that the "grandfather graph" has the following property: it is the unique completion to a transitive graph of a large enough finite subgraph of itself.
2017	pdf arXiv	Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets with Sara Chari, Daniel J. Kelleher, Luke G. Rogers REU. Classical Analysis and ODEs	We prove existence of a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gasket fractals and deduce Gaussian heat kernel bounds in the geodesic metric. Our proof differs from that given by Kigami for the usual Sierpinski gasket in that we show the geodesics are de Rham curves, for which there is an extensive regularity theory.

Notes

Year	Links	Note	Abstract
2018	pdf arXiv	Non-virtually nilpotent groups have infinite conjugacy class quotients with Pooya Vahidi Ferdowsi Group Theory	We offer in this note a self-contained proof of the fact that a finitely generated group is not virtually nilpotent if and only if it has a quotient with the infinite conjugacy class (ICC) propoerty. This proof is a modern presentation of the original proof, by McLain (1956) and Duguid and McLain (1956).