JOSH FRISCH

I am an NSF postdoctoral fellow 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 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 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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Year | Links | Publication | Abstract |
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2021 | pdf arXiv |
Realizations of countable Borel equivalence relations
(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 ${K}_{\sigma}$ 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. |

2019 | pdf arXiv |
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 $\mathbf{p}$ on a finite alphabet $\Sigma $, there exists a $2$-quasi-regular infinite sequence with characters from $\Sigma $ and density of characters equal to $\mathbf{p}$. We also prove that as ${\left|\mathbf{p}\right|}_{\infty}$ 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. |

2019 | pdf arXiv |
Quotients by countable subgroups are hyperfinite
with Forte Shinko
(preprint) Group Theory Dynamical Systems Logic |
We show that for any Polish group $G$ and any countable normal subgroup $\Gamma \u25c1G$, the coset equivalence relation $G/\Gamma $ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite. |

2020 | pdf arXiv |
Lifts of Borel actions on quotient spaces
Accepted to Israel Journal of Mathematics (Special Volume for Benjamin Weiss). 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. |

2020 | pdf arXiv |
A dichotomy for Polish modules
with Forte Shinko
Accepted to Israel Journal of Mathematics. 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. |

2021 | pdf arXiv journal |
Characteristic measures of symbolic dynamical systems
with Omer Tamuz
Accepted to Ergodic Theory and Dynamical Systems. 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
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 | pdf arXiv journal |
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 | pdf arXiv journal |
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 | 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 $c\in 0,\mathrm{log}|A|$ 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 | 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
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. |

( . . . view all publications as an arXiv search result. ) |

Year | Links | Note | Abstract |
---|---|---|---|

2018 | pdf arXiv |
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). |