Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of Gamma-limit groups

A filling subgroup of a finitely generated free group F(X) is a subgroup which does not fix a point in any very small action free action on an R -tree. For the free group of rank two, we construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is filling. In higher ranks, we discuss two types of non-filling subgroups: those contained in loop vertex subgroups and those contained in segment vertex subgroups. We construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is contained in a segment vertex subgroup. We further give a combinatorial algorithm which identifies a certain kind of subgroup contained in a loop vertex subgroup. Finally, we show that the set of filling elements of F(X) is exponentially generic in the sense of Arzhantseva-Ol'shanskiiˇ, refining a result of Kapovich and Lustig.;Let Gamma be a fixed hyperbolic group. The Gamma-limit groups of Sela are exactly the finitely generated, fully residually Gamma groups. We give a new invariant of Gamma-limit groups called Gamma-discriminating complexity and show that the Gamma-discriminating complexity of any Gamma-limit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of Kharlampovich-Myasnikov which states that a Gamma-limit group embeds in an iterated extension of centralizers over Gamma. The result then follows from our proof that if G is an iterated extension of centralizers over Gamma, the G-discriminating complexity of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n.