Limit trees for free group automorphisms universality

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Limit trees for free group automorphisms: universality

Jean Pierre Mutanguha∗

October 16, 2024

Abstract

To any free group automorphism, we associate a universal (cone of) limit tree(s)

with three deﬁning properties: ﬁrst, the tree has a minimal isometric action of the

free group with trivial arc stabilizers; second, there is a unique expanding dilation of

the tree that represents the free group automorphism; and ﬁnally, the loxodromic ele-

ments are exactly the elements that weakly limit to dominating attracting laminations

under forward iteration by the automorphism. So the action on the tree detects the

automorphism’s dominating exponential dynamics.

As a corollary, our previously constructed limit pretree that detects the exponen-

tial dynamics is canonical. We also characterize all very small trees that admit an

expanding homothety representing a given automorphism. In the appendix, we prove

a variation of Feighn–Handel’s recognition theorem for atoroidal outer automorphisms.

Introduction

We previously constructed a limit pretree that detects the exponential dynamics for an

arbitrary free group automorphism [22]. In this sequel, we prove the construction is canon-

ical. This completes the existence and uniqueness theorem for a free group automorphism’s

limit pretree. Recall that if we record all the compact geodesics in an R-tree but forget

their lengths, then the resulting structure is a pretree; brieﬂy, a pretree is a set with a struc-

ture that encodes the notion of closed intervals satisfying certain axioms. Pretrees are the

baseline of our constructions; for instance, (R-)trees will be deﬁned as pretrees with convex

metrics, and pseudotrees as pretrees with a certain hierarchy of convex pseudometrics.

In [22], we motivated the existence and uniqueness theorem of a limit pretree by describ-

ing it as a free group analogue to the Nielsen–Thurston theory for surface homeomorphisms,

which in turn can be seen as the surface analogue to the Jordan canonical form for linear

maps. We now give our own motivation for this result.

∗Email: jpmutang@ias.edu,Web address: https://mutanguha.com

Institute for Advanced Study, Princeton, NJ, USA

MSC Codes 20F65, 20E05, 20E08, 20E36

arXiv:2210.01275v3 [math.GR] 15 Oct 2024

Universal representation of an endomorphism

It feels rather odd to discuss my personal motivation while using the communal “we”;

excuse me as I break this convention a bit for this section. In my doctoral thesis, I extended

Brinkmann’s hyperbolization theorem to mapping tori of free group endomorphisms. This

required studying the dynamics of endomorphisms. Along the way, I proved that injective

endomorphisms have canonical representatives. More precisely, suppose ϕ:F→Fis an

injective endomorphism of a ﬁnitely generated free group; then there is:

1. a minimal simplicial F-action on a simplicial tree Twith trivial edge stabilizers;

2. a ϕ-equivariant expanding embedding f:T→T(unique up to isotopy); and

3. an element in Fis T-elliptic if and only if one of its forward ϕ-iterates is conjugate

to an element in a [ϕ]-periodic free factor of F.

Existence of the limit free splitting (i.e. Twith its F-action) for the outer class [ϕ] was

the core of my thesis (see also [21, Theorem 3.4.5]). Universality follows from bounded

cancellation: any other simplicial tree T′satisfying these three condition will be uniquely

equivariantly isomorphic to T[21, Proposition 3.4.6].

In a way, the limit free splitting detects and ﬁlters the “nonsurjective dynamics” of the

(outer) endomorphism. When ϕ:F→Fis an automorphism, then Tis a singleton and the

free splitting provides no new information. On the other extreme, the F-action on Tcan

be free; in this case, let Γ .

.=F\Tbe the quotient graph. Then the outer endomorphism [ϕ]

is represented by a unique expanding immersion [f]: Γ →Γ and [ϕ] is expansive — such

outer endomorphisms are characterized by the absence of [ϕ]-periodic (conjugacy classes

of) nontrivial free factors [21, Corollary 3.4.8]. The most important thing is that the

expanding immersion [f] has nice dynamics and greatly simpliﬁes the study of expansive

outer endomorphisms.

After completing my thesis, I found myself in a paradoxical situation: I had a better

“understanding” of nonsurjective endomorphisms than automorphisms — the main ob-

stacle to studying the dynamics of nonsurjective endomorphisms was understanding the

dynamics of automorphisms. The na¨ıve expectation (when I started my thesis) had been

that nonsurjective endomorphisms have more complicated dynamics as they are not in-

vertible. The current project was born out of an obligation to correct this imbalance.

Universal representation of an automorphism

What follows is a direct analogue of the above discussion in the setting of automorphisms.

The main theorem of [22] produces an action that detects and ﬁlters the “exponential”

dynamics of an automorphism. Speciﬁcally, suppose ϕ:F→Fis an automorphism of a

ﬁnitely generated free group. Then there is:

1. a minimal rigid F-action on a real pretree Twith trivial arc stabilizers;

2. a ϕ-equivariant “expanding” pretree-automorphism f:T→T; and

3. an element in Fis T-elliptic if and only if it grows polynomially with respect to [ϕ].

The pair of the pretree Tand its rigid F-action is called a (forward) limit pretree for the

outer automorphism [ϕ]. The theorem is stated properly in Chapter III as Theorem III.1.

When [ϕ] is polynomially growing, then the limit pretree is a singleton (and hence unique)

but provides no new information. We are mainly interested in exponentially growing [ϕ]

as their limit pretrees are not singletons. On the other hand, the F-action on a limit

pretree is free if and only if [ϕ] is atoroidal, i.e. there are no [ϕ]-periodic (conjugacy classes

of) nontrivial elements [22, Corollary III.5]. As with expanding immersions and expansive

outer endomorphisms, the expanding “homeomorphism” [f] (on the quotient space F\T)

has dynamics that could facilitate the study of atoroidal outer automorphisms.

Unlike the endomorphism case, uniqueness of limit pretrees requires a more involved

argument. It was remarked in the epilogue of [22] that the only source of indeterminacy

in the existence proof was [22, Proposition III.2]; this proposition is restated in Section I.4

as Proposition I.2 and a proof is sketched in Sections II.1 and II.4. The main result of this

paper is a universal version of the proposition. It can also be thought of as an existence

and uniqueness theorem for an action that detects and ﬁlters the “dominating” exponential

dynamics of an outer automorphism:

Main Theorem (Theorems III.10–III.11).

Let ϕ:F→Fbe an automorphism of a ﬁnitely generated free group and {Adom

j[ϕ]}k

j=1

a (possibly empty) subset of [ϕ]-orbits of dominating attracting laminations for [ϕ].

Then there is:

1. a minimal factored F-tree (Y, Σk

j=1δj)with trivial arc stabilizers;

2. a unique ϕ-equivariant expanding dilation f: (Y, Σk

j=1δj)→(Y, Σk

j=1δj); and

3. for 1≤j≤k, a nontrivial element in Fis δj-loxodromic if and only if its forward

ϕ-iterates have axes that weakly limit to Adom

j[ϕ];

moreover, the factored F-tree (Y, Σk

j=1δj)is unique up to a unique equivariant dilation.

Thus the factored tree (up to rescaling of its factors δj) is a universal construction for outer

automorphisms of free groups, and we call it the complete dominating (resp. topmost) tree if

we consider the whole set of orbits of dominating (resp. topmost) attracting laminations.

As a corollary, the previously constructed limit pretrees are independent of the choices

made in the proof of Theorem III.1, i.e. the limit pretree is canonical (Corollary III.9). Let

us now brieﬂy deﬁne the emphasized terms in the theorem’s statement.

An F-tree is an (R-)tree with an isometric F-action. Informally, an F-tree is factored if

its metric has been equivariantly decomposed as a sum Pk

j=1 δjof pseudometrics. For a fac-

tored F-tree (Y, Σk

j=1δj), an element in Fis δi-loxodromic if it is it is Y-loxodromic and its

axis has positive δi-diameter. An equivariant homeomorphism (T, Σk

j=1dj)→(Y, Σk

j=1δj)

of factored F-trees is a dilation if it is a homothety of each pair of factors djand δj; a

dilation is expanding if each factor-homothety is expanding.

Alamination in Fis a nonempty closed subset in the space of lines in F. A sequence

of lines (e.g. axes) weakly limits to a lamination if some subsequence converges to the

lamination. Any [ϕ] has a ﬁnite set of attracting laminations which is empty if and only

if [ϕ] is polynomially growing; this set is partially ordered by inclusion and has an order-

preserving [ϕ]-action. The maximal elements of the partial order are called topmost. An

attracting lamination Afor [ϕ] has an associated stretch factor λ(A); it is dominating if

any distinct attracting lamination A′for [ϕ] containing Ahas a strictly smaller stretch

factor λ(A′)< λ(A). Topmost attracting laminations are vacuously dominating; moreover,

the [ϕ]-action permutes the dominating attracting laminations.

Remark. If one considers a subset {Atop

j[ϕ]}k

j=1 of [ϕ]-orbits of topmost attracting lam-

inations, then we prove the topmost tree has the additional property that its factor-

pseudometrics are pairwise mutually singular : for each i, there is an element that is

δi-loxodromic but δj-elliptic for j̸=i(see Section III.4). We highlight this feature by

using the notation (Y, ⊕k

j=1δj) for topmost trees.

Some applications of universal representations. Fix an automorphism ϕ:F→F;

since [ϕ] has a unique equivariant dilation class [Y, Σk

j=1δj] of complete dominating limit

trees, any invariant of the class is automatically an invariant of [ϕ]. For instance, the

Gaboriau–Levitt index i(Y) (as deﬁned in [11, Chapter III]) is the dominating forward

index for [ϕ]. In fact, since the limit pretree Tfor [ϕ] is canonical, its index i(T) (deﬁned

in [22, Appendix A]) is the exponential (forward) index for [ϕ]; when [ϕ] is atoroidal, the

index i(T) is closely related to the Gaboriau–Jaeger–Levitt–Lustig index for [ϕ] deﬁned

in [10, Section 6]. Each factor δjhas an associated F-tree (Ydom

j, δj); the pairing of δj

with the orbit of dominating attracting lamination Adom

j[ϕ] means i(Ydom

j) is an index

for Adom

j[ϕ] respectively.

Our main application is a characterization of minimal F-trees with ϕ-equivariant ex-

panding homotheties:

Main Corollary (Theorem V.3).

Let ϕ:F→Fbe an automorphism and (Y, δ)a minimal very small F-tree. The F-

tree (Y, δ)admits a ϕ-equivariant expanding homothety if and only if it is equivariantly

isometric to the dominating tree for [ϕ]with respect to a subset of [ϕ]-orbits of dominating

attracting laminations with the same stretch factor.

In the appendix, we prove a variation of Feighn–Handel’s recognition theorem for

atoroidal outer automorphisms.

Some historical context

This paper continues Gaboriau–Levitt–Lustig’s philosophy of prioritizing limit trees in

their alternative proof of the Scott conjecture [12]. In particular, our paper relies only

on the existence of irreducible train tracks [4, Section 1] but none of the typical splitting

paths analysis of relative train tracks [3, 9]. Zlil Sela gave another dendrogical proof the

conjecture (now Bestvina–Handel’s theorem) that used Rips’ theorem in place of train

track technology [25]. Fr´ed´eric Paulin gave yet another dendrological proof that avoids

both train tracks and Rips’ theorem [23].

About the same time, Bestvina–Fieghn–Handel used train tracks and trees to prove fully

irreducible (outer) automorphisms have universal limit trees [2]. They used this to give a

short dendrological proof of a special case of the Tits Alternative for Out(F); their later

proof of the general case was much more involved due to the lack of such a universal limit

construction [3]. Universal limit trees have been indispensable for studying fully irreducible

automorphisms. In principle, a universal construction of limit trees for all automorphisms

would lead to a dendrological proof of the Tits alternative and extend much of the theory

for fully irreducible automorphisms to arbitrary automorphisms. Speaking of dendrological

proofs of the Tits alternative, we mention that Camille Horbez gave such a proof with a

very diﬀerent approach [15].

Continuing the work started in [3], Feighn–Handel deﬁned and proved the existence

of completely split relative train tracks (CTs) in [9, Section 4]; they use CTs to charac-

terize abelian subgroup of Out(F) [8]. The main obstacle when working with topological

representatives is that they are not canonical, which can make deﬁning invariants of the

outer automorphism quite technical. This is the diﬃculty that we had to deal with in this

paper; however, now that it is done, we can use our new universal representatives to deﬁne

other invariants rather easily. A minor inconvenience when working with CTs is that they

are only proven to exist for some (uniform) iterate of the outer automorphism; we were

very careful (perhaps to a fault) in this paper to ensure our universal representatives exist

for all outer automorphisms. Finally, a subtle advantage to our approach is that we ﬁnd

universal representatives for automorphisms and not just outer automorphisms!

In a sequel to [25], Sela used limit trees and Rips’ theorem to give a canonical hierar-

chical decomposition of the free group Fthat is invariant under a given atoroidal auto-

morphism [24]. This second paper was never published and a third announced paper that

extends the canonical decomposition to arbitrary automorphisms was never released even

as a preprint (as far as we know). We remark that the limit trees used in that paper were

not (or rather, were never proven to be) canonical/universal. Perhaps, one could combine

Sela’s canonical decomposition with Bestvina–Feighn–Handel’s work to give a universal

construction of limit trees for atoroidal automorphisms — our approach is independent

of Sela’s work and applies more generally to exponentially growing automorphisms. Con-

versely, we suspect that a careful study of the structure of our topmost trees might recover

Sela’s canonical hierarchical decomposition.

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摘要：

Limittreesforfreegroupautomorphisms:universalityJeanPierreMutanguha∗October16,2024AbstractToanyfreegroupautomorphism,weassociateauniversal(coneof)limittree(s)withthreedefiningproperties:first,thetreehasaminimalisometricactionofthefreegroupwithtrivialarcstabilizers;second,thereisauniqueexpandingdilat...

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Limit trees for free group automorphisms universality

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