Words have bounded width in $\operatorname{SL}(n, \mathbb{Z})$. (July 2019)

• Record Type: Journal Article
• Authors: Avni, Nir; Meiri, Chen
• Is Part Of: Compositio mathematica. Volume 155:Number 7(2019)
• Page Start: 1245
• Page End: 1258
• Publication Date: 2019-07
• Subjects: 20H05, -- 11E57 (primary); arithmetic groups, -- word width; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X19007334
• Languages: English
• ISSNs: 0010-437X

Abstract

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.