Currents relative to a malnormal subgroup system

Abstract

This paper introduces a new topological space associated with a nonabelian free group F n of rank n and a malnormal subgroup system A of F n , called the space of currents relative to A, which are F n -invariant measures on an appropriate subspace of the double boundary of F n . The extension from free factor systems as considered by Gupta to malnormal subgroup systems is necessary in order to fully study the growth under iteration of outer automorphisms of F n , and requires the introduction of new techniques on cylinders. We in particular prove that currents associated with elements of F n which are not contained in a conjugate of a subgroup of A are dense in the space of currents relative to A. 1

North-South type dynamics of relative atoroidal automorphisms of free groups on a relative space of currents

“…By Lemma 3.28, the set consisting in elements Cpγq with γ P PpF 1 q covers B 2 pA, F 1 q. Thus, by [Gue1,Lemma 3.2], it suffices to prove that for every projective current η P PCurrpSupppη e qq, we can associate a function r η : P A pF 1 q Ñ R such that for every γ P P A pF 1 q, we have piq 0 ď r ηpγq ă 8; piiq r ηpγq " σ γ ´1 ; piiiq r ηpγq " ř ePE σ γe , where E is the subset of EG 1 consisting in all edges that are incident to the endpoints of γ and distinct from the inverse of the last edge of γ.…”

“…This fact is sufficient to prove that lim mÑ8 φ m prµsq P ∆ `pφq (see Lemma 5.20). We then conclude the proof using the density of currents associated with nonperipheral elements in F n proved in [Gue1]. Theorem 1.3 is then proved in Section 6 using a combination of Theorem 1.2 and the description of the space K P G .…”

“…Lemma 2.14. [Gue1,Lemma 3.3] Let n ě 3 and let A be a malnormal subgroup system of F n . The space PCurrpF n , Aq is compact.…”

“…Moreover, we have Polypφq " Ť r i"1 Ť gPFn gH i g ´1. In [Gue1], we construct a compact, metrizable space, called the space of projectivised currents relative to Apφq, denoted by PCurrpF n , Apφqq, which is the space of projectivised Radon measures on the double boundary of F n relative to Apφq, equipped with the weak-star topology (see Section 2.4 for precise definitions). In [Gue1], we proved that the set of currents associated with Apφq-nonperipheral conjugacy classes of elements of g of F n , that is, such that g is not contained in the conjugacy class of some H i with i P t1, .…”

Polynomial growth and subgroups of $\mathrm{Out}(F_{\tt n})$

“…If the extension F i´1 ď F i is nonsporadic (see the definition in Section 2.1) then the construction of φ i from φ i´1 follows from the works of Handel-Mosher [HM], and Clay-Uyanik [CU2]. If the extension F i´1 ď F i is sporadic, the construction of φ i relies on the action of H on some natural (compact, metrizable) space that we introduced in [Gue1]. This space is called the space of currents relative to PolypH| F i´1 q, denoted by PCurrpF n , PolypH| F i´1 qq.…”