A Short Proof of the Grigorchuk-Cohen Cogrowth Theorem

Abstract

Let G be a group generated by g\,...,gr ■ There are exactly 2r(2r -l)"_l reduced words in g\,...,gr of length n .Part of them, say y" represents identity element of G .Let y = lim sup y" .We give a short proof of the theorem of Grigorchuk and Cohen which states that G is amenable if and only if y = 2r -12 .Moreover we derive some new properties of the generating function ¿^Lynz" ■

Locations

• Proceedings of the American Mathematical Society - View - PDF