## Bio

JOSH FRISCH

I am a 4th year graduate student at Caltech working under Alekos Kechris and Omer Tamuz. My main research interests are in the interaction between group theory and dynamics.

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

Some particular topics I have enjoy thinking about include Entropy Theory, 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.

## Publications

2019
Quasi-Regular Sequences

(preprint)
Combinatorics
Let $\Sigma$ be a countable alphabet. For $r\ge 1$, an infinite sequence $s$ with characters from $\Sigma$ is called $r$-quasi-regular, if for each $\sigma \mathrm{in\Sigma }$ the ratio of the longest to shortest interval between consecutive occurrences of $\sigma$ 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 $\Sigma$, there exists a $2$-quasi-regular infinite sequence with characters from $\Sigma$ 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.
2019
Quotients by countable subgroups are hyperfinite

(preprint)
Group Theory
Dynamical Systems
Logic
We show that for any Polish group $G$ and any countable normal subgroup $\Gamma ◁G$, the coset equivalence relation $G/\Gamma$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
2019
Characteristic measures of symbolic dynamical systems

(preprint)
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.
2019
Choquet-Deny groups and the infinite conjugacy class property

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 $\mu$ on $G$ it holds that all bounded $\mu$-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
Strong amenability and the infinite conjugacy class property

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
Normal amenable subgroups of the automorphism group of the full shift

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
Symbolic dynamics on amenable groups: the entropy of generic shifts

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 $c\in 0,\mathrm{log}|A|$ 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 ${G}_{\delta }$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.
2016
Transitive graphs uniquely determined by their local structure

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
Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets

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

2018
Non-virtually nilpotent groups have infinite conjugacy class quotients

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).