Genericity of Filling Elements

TL;DR

This paper proves that filling elements in free groups are exponentially generic, provides an algebraic criterion for filling, and identifies a large subset with a linear-time membership problem.

Contribution

It establishes the exponential genericity of filling elements, offers an algebraic condition for filling, and finds a large subset with an efficiently solvable membership problem.

Findings

Filling elements are exponentially generic in free groups.

An algebraic criterion for identifying filling elements.

Existence of a large subset of filling elements with linear-time membership testing.

Abstract

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 action of F(X) on an $\mathbb{R}$-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'shanskii. 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.