Approximate subgroups of linear groups

Abstract

We establish various results on the structure of approximate subgroups in linear groups such as SL_n(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(F_q) which generates the group must be either very small or else nearly all of SL_n(F_q). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over a finite field k. In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.

48 pages

References in corpus (5)

• A Strong Tits Alternative
• Expansion properties of finite simple groups
• Growth in SL_3(Z/pZ)
• Growth in finite simple groups of Lie type
• Suzuki groups as expanders

Cited by in corpus (6)

• Growth in finite simple groups of Lie type of bounded rank
• A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups
• Suzuki groups as expanders
• Expander Graphs in Pure and Applied Mathematics
• Sieve methods in group theory I: Powers in Linear groups
• Strongly dense free subgroups of semisimple algebraic groups