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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" ■ Let G be a finitely generated discrete group. Consider G as an epimorphic image n : Fr -► G of the free group Fr on r generators. Thus G is isomorphic to the quotient group Fr/N where N = ker n is a normal subgroup of Fr. Once we fix a set of free generators in Fr we introduce |x| the length of the word x in Fr with respect to the generators and their inverses. Let Nn = {x G N: \x\ = n} , 1 In yn = card Nn and y = lim sup yn' which is called the growth exponent of N is Fr with respect to the fixed set of free generators in Fr. Because there are exactly 2r(2r -l)"~ elements of length n in Fr, y <2r -I. Grigorchuk [2] and Cohen [ 1 ] have shown that a group G is amenable if and only if y attains maximal possible value, i.e. y = 2r -1 . We propose a new rather simple proof without any estimates which allows us to draw out new information on the behaviour of the generating function N(z) = J^ynz".As in [1] and [2] we will base our proof on a characterization of discrete amenable groups given by Kesten [3].
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" ■ Let G be a finitely generated discrete group. Consider G as an epimorphic image n : Fr -► G of the free group Fr on r generators. Thus G is isomorphic to the quotient group Fr/N where N = ker n is a normal subgroup of Fr. Once we fix a set of free generators in Fr we introduce |x| the length of the word x in Fr with respect to the generators and their inverses. Let Nn = {x G N: \x\ = n} , 1 In yn = card Nn and y = lim sup yn' which is called the growth exponent of N is Fr with respect to the fixed set of free generators in Fr. Because there are exactly 2r(2r -l)"~ elements of length n in Fr, y <2r -I. Grigorchuk [2] and Cohen [ 1 ] have shown that a group G is amenable if and only if y attains maximal possible value, i.e. y = 2r -1 . We propose a new rather simple proof without any estimates which allows us to draw out new information on the behaviour of the generating function N(z) = J^ynz".As in [1] and [2] we will base our proof on a characterization of discrete amenable groups given by Kesten [3].
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