# Doron Puder

Professor,

School of Mathematical Sciences,

Tel Aviv University

Room 235

Office hours: By appointment

Office phone: +972-3-640-8235

Email: doronpuder at gmail

In the year 2024-2025 I'm on sabbatical in the School of Mathematics, at the Institute for Advanced Study in Princeton.

ERC Project: Word Measures in Groups and Random Cayley Graphs

### Research Interests:

I study group theory, and especially combinatorial, geometric and probabilistic group theory. Many of the questions in my research involve word measures on groups: measures induced by word maps. This involves the study of free groups and their automorphism group, as well as certain aspects of the representation theory of the groups on which the word measures are induced. In the background there are problems about expansion and spectral gaps of random objects (random graphs, random surfaces, random Cayley graphs and so on).

Groups & Dynamics Seminar: Thursdays at 11:00

## Preprints:

The Ring of Stable Characters over GL.(q). With Danielle Ersnt-West and Yotam Shomroni. [arXiv]

An Extension of the Cogrowth Formula to Arbitrary Subsets of the Tree. [arXiv]

Stable Invariants and Their Role in Word Measures on Groups. With Yotam Shomroni. [arXiv]

On the Aldous-Caputo Spectral Gap Conjecture for Hypergraphs. With Gil Alon and Gady Kozma. [arXiv]

## Papers:

Word Measures on GLn(q) and Free Group Algebras. With Danielle Ernst-West and Matan Seidel. [journal, arXiv]

Algebra & Number Theory, 18(11), 2047-2090, 2024Local Statistics of Random Permutations from Free Products. With Tomer Zimhoni. [journal, arXiv]

International Mathematics Research Notices (IMRN), 2024 (5), 4242-4300, 2024The Asymptotic Statistics of Random Covering Surfaces. With Michael Magee. [journal, arXiv, mini-course (part I, part II, part III, part IV)]

Forum of Mathematics, Pi, 11:e15, 1-51, 2023Matrix Group Integrals, Surfaces, and Mapping Class Groups II: O(n) and Sp(n). With Michael Magee. [journal, arXiv]

Mathematische Annalen, 388, 1437--1494, 2024Core Surfaces. With Michael Magee. [journal, arXiv]

Geometriae Dedicata, 216:46, 2022A Random Cover of a Compact Hyperbolic Surface Has Relative Spectral Gap 3/16-ε. With Michael Magee and Frédéric Naud. [journal, arXiv]

Geometric and Functional Analysis (GAFA), 32, 595--661, 2022Word Measures on Symmetric Groups. With Liam Hanany. [journal, arXiv]

International Mathematics Research Notices (IMRN), 2023 (11), 9221--9297, 2023

Extended abstract in proceedings of FPSAC 2020 [journal, PDF]A Note on the Trace Method for Random Regular Graphs. With Joel Friedman. [journal, arXiv]

Addendum to Expansion of Random Graphs: New Proofs, New Results.

Israel Journal of Mathematics, 256, 269--282, 2023Some Orbits of Free Words that are Determined by Measures on Finite Groups. With Liam Hanany and Chen Meiri. [journal, arXiv]

Journal of Algebra 555, 305--324, 2020

Aldous' Spectral Gap Conjecture for Normal Sets. With Ori Parzanchevski. [journal, arXiv]

Transactions of the American Mathematical Society 373 (10), 7067--7086, 2020

Surface Words are Determined by Word Measures on Groups. With Michael Magee. [journal, arXiv]

Israel Journal of Mathematics 241, 749--774, 2021

Automorphism-Invariant Positive Definite Functions on Free Groups. With Benoît Collins and Michael Magee. [arXiv]

Proceedings of the 27th International Conference in Operator Theory (OT27), 115--137, 2020

Matrix Group Integrals, Surfaces, and Mapping Class Groups I: U(n). With Michael Magee. [journal, arXiv]

Inventiones Mathematicae 218 (2), 341--411, 2019

The Markoff Group of Transformations in Prime and Composite Moduli. With Chen Meiri and appendix by Dan Carmon. [journal, arXiv]

Duke Mathematical Journal 167 (14), 2679--2720, 2018

Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n). With Alexandru Nica and Ian P. Goulden. [journal, arXiv]

International Mathematics Research Notices (IMRN) 2020 (4), 983--1034, 2020

Ramanujan Coverings of Graphs. With Chris Hall and Will Sawin. [journal, arXiv, STOC (extended abstract)]

Advances in Mathematics 323 ,367--410, 2018

STOC 2016, 48th annual ACM SIGACT Symposium in the Theory of Computing, 533--541, 2016. [Extended abstract]

Expansion of Random Graphs: New Proofs, New Results. [journal, arXiv]

Inventiones Mathematicae 201 (3), 845--908, 2015

Addendum with Joel Friedman: [arXiv]

Measure Preserving Words are Primitive. With Ori Parzanchevski. [journal, arXiv, 50-minute-talk, 4-hours-talk (part I, part II)]

Journal of the American Mathematical Society 28 (1), 63--97, 2015

Growth of Primitive Elements in Free Groups. With Conan Wu. [journal, arXiv, The World of Groups - Question F19 ]

Journal of the London Mathematical Society 90 (1), 89--104, 2014

Stalling Graphs, Algebraic Extensions and Primitive Elements in F2. With Ori Parzanchevski. [journal, arXiv]

Mathematical Proceedings of the Cambridge Philosophical Society 157 (1), 1--11, 2014

Primitive Words, Free Factors and Measure Preservation. [journal, arXiv]

Israel Journal of Mathematics 201 (1), 25--73, 2014

Word Maps and Spectra of Random Graph Lifts. With Nati Linial. [journal, arXiv]

Random Structures and Algorithms 37 (1) 100--135, 2010

## Not for publication:

Word Measures on Unitary Groups. With Michael Magee. [arXiv]

The results of this preprint have been extended and generalized in the paper "Matrix Group Integrals, Surfaces, and Mapping Class Groups I: U(n)" appearing above. However, this preprint contains useful background, examples and a somewhat different approach in the proofs.

## Book:

Algebraic Structures: Groups, Rings and Fields (Textbook in Hebrew). With Ehud de Shalit and Alex Lubotzky.

מבנים אלגבריים: חבורות, חוגים ושדות

Errata file (please email me about any other mistakes and typos in the book)

## Post-docs:

Michal Buran (2020-2021)

Daniele Garzoni (2021-2023)

Noam Kolodner (2022- 2024)

## Phd Students:

Matan Seidel

Danielle Ernst-West

Niv Levhari

## Master Students:

Danielle Ernst-West (completed 2019): Word measures on GLn(Fq)

Ziv Greenhut (completed 2019): A generation criterion for subsets of special linear groups over finite fields

Gal Ordo (completed 2020): Word measures on the signed symmetric group

Liam Hanany (completed 2021): Word measures on symmetric groups

Yaron Brodsky (completed 2022): Word Measures on Unitary Groups: Improved Bounds for Small Representations

Tomer Zimhoni (completed 2023): Local statistics of random permutations from free products

Yotam Shomroni (completed 2023): Word Measures on Wreath Products I

Dror Frid

Noam Ta-Shma

## Mini-courses in conferences:

In Thematic Program on Randomness and Geometry, Toronto, 29 April - 10 May, 2024

Title: Random covers of graphs and surfacesIn Probabilistic Methods in Negative Curvature, Bangalore, February-March 2023

Title: Random covers of Hyperbolic SurfacesIn Combinatorics of finite-index subgroups, Paris, November 2022

Title: The asymptotic statistics of random covering surfacesIn Paroles, Paroles: A Summer School on Words, Haifa, July 2022

Title: Word measures on symmetric and other groupsIn Group Actions, Geometry and Dynamics, Oholo, July 2022

Title: Free groups, surfaces and linear programmingIn Laplacians on Random Hyperbolic Surfaces and on Random Graphs, Evanston, May 2022

Title: The spectrum of random graphsIn Recent trends in the study of expanders and high dimensional expanders, Rehovot, February 2017

Title: New trends in expanders based on the method of interlacing polynomials

## Advanced Classes Taught:

Random Walks on Groups - Fall 2016

Surfaces: Graphs, Groups and Applications - Spring 2018

Topics in Combinatorial Group Theory - Fall 2018

Representations of Finite Groups - Spring 2020

Introduction to Algebraic Topology - Fall 2020

Surfaces: Graphs, Groups and Applications - Fall 2021

Advanced seminar in geometric group theory - Spring 2022

## Brief CV:

2024-2025: Von-Neumann fellow, School of Mathematics, IAS, Princeton

2023 - present: Professor, School of Mathematical Sciences, Tel Aviv University

2019-2023: Associate Professor, School of Mathematical Sciences, Tel Aviv University

2016-2019: Senior Lecturer, School of Mathematical Sciences, Tel Aviv University

2014-2016: Member, School of Mathematics, IAS, Princeton

2010-2014: PhD, Einstein Institute of Mathematics, at the Hebrew University of Jerusalem, supervised by Nati Linial.