Counting lattices in products of trees

Nir Lazarovich; Ivan Levcovitz; Alex Margolis

doi:10.4171/CMH/559

Counting lattices in products of trees

Nir Lazarovich, Ivan Levcovitz, Alex Margolis

Mathematics

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

A BMW group of degree .m; n/ is a group that acts simply transitively on vertices of the product of two regular trees of degrees m and n. We show that the number of commensurability classes of BMW groups of degree .m; n/ is bounded between .mn/α^mn and .mn/β^mn for some 0 < α < β. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree .m; n/ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.

Original language	English
Pages (from-to)	597-630
Number of pages	34
Journal	Commentarii Mathematici Helvetici
Volume	98
Issue number	3
DOIs	https://doi.org/10.4171/CMH/559
State	Published - 2023

Keywords

CAT(0) square complexes
Lattices
product of trees
random complexes
simple groups

All Science Journal Classification (ASJC) codes

General Mathematics

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10.4171/CMH/559

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title = "Counting lattices in products of trees",

abstract = "A BMW group of degree .m; n/ is a group that acts simply transitively on vertices of the product of two regular trees of degrees m and n. We show that the number of commensurability classes of BMW groups of degree .m; n/ is bounded between .mn/αmn and .mn/βmn for some 0 < α < β. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree .m; n/ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.",

keywords = "CAT(0) square complexes, Lattices, product of trees, random complexes, simple groups",

author = "Nir Lazarovich and Ivan Levcovitz and Alex Margolis",

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AU - Lazarovich, Nir

AU - Levcovitz, Ivan

AU - Margolis, Alex

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N2 - A BMW group of degree .m; n/ is a group that acts simply transitively on vertices of the product of two regular trees of degrees m and n. We show that the number of commensurability classes of BMW groups of degree .m; n/ is bounded between .mn/αmn and .mn/βmn for some 0 < α < β. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree .m; n/ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.

AB - A BMW group of degree .m; n/ is a group that acts simply transitively on vertices of the product of two regular trees of degrees m and n. We show that the number of commensurability classes of BMW groups of degree .m; n/ is bounded between .mn/αmn and .mn/βmn for some 0 < α < β. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree .m; n/ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.

KW - CAT(0) square complexes

KW - Lattices

KW - product of trees

KW - random complexes

KW - simple groups

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EP - 630

JO - Commentarii Mathematici Helvetici

JF - Commentarii Mathematici Helvetici

IS - 3

ER -

Counting lattices in products of trees

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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