OPERATOR GROWTH FUNCTIONS OF DISCRETE GROUPS.pdf

OPERATOR GROWTH FUNCTIONS OF DISCRETE GROUPSROSTISLAV GRIGORCHUK AND TATIANA NAGNIBEDAAbstract. A generalizationof the growth functions of nitelygeneratedgroupsis introduced, namely the growth functionswith operator coecients g2Ggzjgj .The questions of rationality and convergence of such series are discussed. Theoperator growth functions of surface groups are explicitly computed. The op-erator geodesic growth functions are also studied.1. IntroductionConsider a nitely generated discrete group G with a nite system of semigroupgenerators S = fs1; :::; skg; 1 =2 S, and the corresponding length metric. The growthfunction of G with respect to S is the power seriesf(z) = 1Xn=0anzn 2 Z[z] ;where an is the number of elements of G of length n. The growth functions ofdiscrete groups have been studied by many authors. Let us mention the compu-tations of the growth functions of Coxeter groups ([33], [6], [26], [34], [7]), surfacegroups and fuchsian groups ([7], [34], [11]), Heisenberg and Nil groups ([29], [5]).In all these examples the growth functions are rational. They are also rational forvirtually abelian groups ([4],[21]) and for hyperbolic groups ([8], [14], [12]). Thereexist however nilpotent groups and systems of generators in them, such that thecorresponding growth functions are not rational ([15], [31]).In this paper we study the rationality and the methods of computation of a nat-ural generalization of the growth function. We dene the complete growth functionof the group G with respect to the system of generators S asF (z) = Xg2G gzjgj = 1Xn=0Anzn 2 Z[G][z] ;where An is the sum of all elements of G of length n viewed as an element of thegroup ring, i.e., An = Xg2G;jgj=n g 2 Z[G] ;Key words and phrases. Growth function, Cayley graph, formal power series, hyperbolic group,surface group.Steklov Institute of Mathematics, Vavilova Str. 42, Moscow 117966, Russia, e-mail: grig-orch@alesia.ips.ras.ru, grigorch@mian.su.Section de mathematiques, Universite de Gen