RANDOM OPERATOR APPROACH FOR WORD ENUMERATION IN BRAID-GROUPS

We investigate analytically the problem of enumeration of nonequivalen t primitive words in the locally free, LFn, and braid groups B-n, for n much greater than 1 by analysing the random word statistics and targ et spaces on these groups. We develop a 'symbolic dynamics' method for exact word enumeration in locally free groups and give arguments in s upport of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation r elations. We touch briefly the connection of these problems with conve ntional random operator theory, localization phenomena and statistics of systems with quenched disorder. We also discuss the relation of par ticular problems of random operator theory to the theory of modular fu nctions.