Random operator approach for word enumeration in braid groups

Abstract

We investigate analytically the problem of enumeration of nonequivalent primitive words in the locally free, and braid groups for by analysing the random word statistics and target spaces on these groups. We develop a `symbolic dynamics' method for exact word enumeration in locally free groups and give arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We touch briefly the connection of these problems with conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. We also discuss the relation of particular problems of random operator theory to the theory of modular functions.