An element of a finitely generated non-Abelian free group F (X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F (X) on an R-tree. We give a proof that the set of filling elements of F (X) is exponentially F (X)-generic in the sense of Arzhantseva and Ol'shanskiȋ. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F (X)generic subset of filling elements whose membership problem is solvable in linear time.
We use a recent result of Alexander and Nishinaka to show that if G is a nonelementary torsion-free hyperbolic group and R is a countable domain, then the group ring RG is primitive. This implies that the group ring KG of any non-elementary torsion-free hyperbolic group G over a field K is primitive.
Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We give a new invariant of Γ-limit groups called Γ-discriminating complexity and show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of Kharlampovich-Myasnikov which states that a Γ-limit group embeds in an iterated extension of centralizers over Γ. The result then follows from our proof that if G is an iterated extension of centralizers over Γ, the G-discriminating complexity of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n.
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