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Words have bounded width in $\operatorname{SL}(n,\mathbb{Z})$

Part of: Forms and linear algebraic groups Other groups of matrices

Published online by Cambridge University Press:  13 June 2019

Nir Avni  and
Chen Meiri
Nir Avni
Affiliation:
Department of Mathematics, Northwestern University, Evanston IL, USA email avni.nir@gmail.com
Chen Meiri
Affiliation:
Department of Mathematics, Technion, Haifa, Israel email chenm@technion.ac.il
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Abstract

We prove two results about the width of words in $\operatorname{SL}_{n}(\mathbb{Z})$. The first is that, for every $n\geqslant 3$, there is a constant $C(n)$ such that the width of any word in $\operatorname{SL}_{n}(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $\operatorname{SL}_{n}(\mathbb{Z})$ is at most 87.

Keywords

arithmetic groupsword width

MSC classification

Primary: 20H05: Unimodular groups, congruence subgroups 11E57: Classical groups
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 7 , July 2019 , pp. 1245 - 1258
Copyright
© The Authors 2019 

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References

Borel, A., On free subgroups of semisimple groups , Enseign. Math. (2) 29 (1983), 151–164.Google Scholar
Dennis, R. K. and Vaserstein, L. N., On a question of M. Newman on the number of commutators , J. Algebra 118 (1988), 150–161.Google Scholar
Larsen, M. and Shalev, A., Word maps and Waring type problems , J. Amer. Math. Soc. 22 (2009), 437–466.10.1090/S0894-0347-08-00615-2Google Scholar
Liebeck, M. and Shalev, A., Diameters of finite simple groups: sharp bounds and applications , Ann. of Math. (2) 154 (2001), 383–406.Google Scholar
Lubotzky, A., Images of word maps in finite simple groups , Glasg. Math. J. 56 (2014), 465–469.Google Scholar
Mennicke, J. L., Finite factor groups of the unimodular group , Ann. of Math. (2) 81 (1965), 31–37.10.2307/1970380Google Scholar
Myasnikov, A. and Nikolaev, A., Verbal subgroups of hyperbolic groups have infinite width , J. Lond. Math. Soc. (2) 90 (2014), 573–591.Google Scholar
Morgan, A. V., Rapinchuk, A. S. and Sury, B., Bounded generation of SL2 over rings of S-integers with infinitely many units , Algebra Number Theory 12 (2018), 1949–1974.Google Scholar
Segal, D., Words: notes on verbal width in groups, London Mathematical Society Lecture Note Series, vol. 361 (Cambridge University Press, Cambridge, 2009).Google Scholar
Thom, A., Convergent sequences in discrete groups , Canad. Math. Bull. 56 (2013), 424–433.10.4153/CMB-2011-155-3Google Scholar
Tits, J., Systèmes générateurs de groupes de congruence , C. R. Acad. Sci. Paris Ser. A-B 283 (1976), A693–A695.Google Scholar
Witte Morris, D., Bounded generation of SL(n, A) (after D. Carter, G. Keller and E. Paige) , New York J. Math. 13 (2007), 383–421.Google Scholar