REGULAR COMBINGS, NONPOSITIVE CURVATURE AND THE QUASICONVEXITY OF ABELIAN SUBGROUPS

We study the geometry of abelian subgroups relative to combings (norma l forms) of automatic groups and groups which act properly and cocompa ctly on spaces of nonpositive curvature. We prove that in general comb ings which arise in the latter context are incompatible (in a sense wh ich we make precise) with regular combings, the fundamental objects of automatic group theory.