# Seminars & Colloquia Calendar

## Arithmetic and quasi-arithmetic hyperbolic reflection groups

#### Nikolay Bogachev (Skoltech & MIPT)

Location: ** Zoom Link: https://rutgers.zoom.us/j/95245984714?pwd=cXJXTldjUGpxdUk5WW9GMVhaREZ6UT09**

Date & time: Tuesday, 22 September 2020 at 2:00PM - 3:00PM

Abstract: In 1967, Vinberg started a systematic study of hyperbolic reflection groups. In particular, he showed that Coxeter polytopes are natural fundamental domains of hyperbolic reflection groups and developed practically efficient methods that allow to determine compactness or volume finiteness of a given Coxeter polytope by looking at its Coxeter diagram. He also proved a (quasi-)arithmeticity criterion for hyperbolic lattices generated by reflections. In 1981, Vinberg showed that there are no compact Coxeter polytopes in hyperbolic spaces \(H^n\) for \(n>29\). Also, he showed that there are no arithmetic hyperbolic reflection groups \(H^n\) for \(n>29\), either. Due to the results of Nikulin (2007) and Agol, Belolipetsky, Storm, and Whyte (2008) it became known that there are only finitely many maximal arithmetic hyperbolic reflection groups in all dimensions. These results give hope that maximal arithmetic hyperbolic reflection groups can be classified.

A very interesting moment is that compact Coxeter polytopes are known only up to \(H^8\), and in \(H^7\) and \(H^8\) all the known examples are arithmetic. Thus, besides the problem of classification of arithmetic hyperbolic reflection groups (which remains open since 1970-80s) we have another very natural question (which is again open since 1980s): do there exist compact (both arithmetic and non-arithmetic) hyperbolic Coxeter polytopes in \(H^n\) for \(n>8\) ?

The talk will be devoted to the discussion of these two related problems. One part of the talk is based on the recent preprint https://arxiv.org/abs/2003.11944v2 where some new geometric classification method is described. The second part is based on a joint work with Alexander Kolpakov https://arxiv.org/abs/2002.11445v2 where we prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a few illustrative examples.

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

P. Gupta, X.Huang and J. Song Organizer's webpage

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

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