CLASS NUMBER FOR PSEUDO-ANOSOVS FRANC OIS DAHMANI AND MAHAN MJ Abstract. Given two automorphisms of a group G one is interested in knowing

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CLASS NUMBER FOR PSEUDO-ANOSOVS

FRANC¸ OIS DAHMANI AND MAHAN MJ

Abstract. Given two automorphisms of a group G, one is interested in knowing

whether they are conjugate in the automorphism group of G, or in the abstract

commensurator of G, and how these two properties may diﬀer. When Gis the

fundamental group of a closed orientable surface, we present a uniform ﬁniteness

theorem for the class of pseudo-Anosov automorphisms. We present an explicit

example of a commensurably conjugate pair of pseudo-Anosov automorphisms of

a genus 3 surface, that are not conjugate in the mapping class group, and we also

show that inﬁnitely many pairwise non-commuting pseud-Anosov automorphisms

have class number equal to one. In the appendix, we brieﬂy survey the Latimer-

MacDuﬀee theorem that addresses the case of automorphisms of Zn, with a point

of view that is suited to an analogy with surface group automorphisms.

1. Introduction: commensurated conjugacy

When Gis a group (or any structure), a natural problem is to classify the conjugacy

classes in its automorphism group Aut pGq. A familiar example is G“Zn, for which

Aut pGq » GLnpZq.

The abstract commensurator Comm pGqof Gis the group of equivalence classes

of isomorphisms between two ﬁnite index subgroups of G, where two such auto-

morphisms ϕ1, ϕ2are declared to be equivalent if they agree on further ﬁnite index

subgroups.

There is a natural homomorphism from Aut pGqto Comm pGq. It need not be

injective, but in many cases of interest it is. In general, Comm pGqis much larger

than the image of Aut pGq. In our familiar example, Comm pZnq » GLnpQq. Of

course, with linear algebra, conjugation in GLnpQqis more easily understood than

in GLnpZq. (See [27] for instance for more details on commensurators in solvable Lie

groups.)

Both authors warmly thank the Institut Henri Poincar´e (IHP) for hosting the program ”Groups

Acting on Fractals” in Spring 2022, the Institut des Hautes Etudes Scientiﬁques for the ﬁrst author’s

visiting Carmin position, and the Centre de Mathematiques Laurent Schwartz. This work was

supported by LabEx CARMIN, ANR-10-LABX-59-01. FD is supported by ANR-22-CE40-0004

GoFR. MM is supported by the Department of Atomic Energy, Government of India, under project

no.12-R&D-TFR-5.01-0500, by an endowment of the Infosys Foundation. and by a DST JC Bose

Fellowship.

arXiv:2210.12824v4 [math.GR] 31 Aug 2024

2 FRANC¸ OIS DAHMANI AND MAHAN MJ

For a group G, and ϕPAut pGq, let rϕsdenote the image of ϕin Comm pGq. We

deﬁne the commensurated-conjugacy class of ϕto be the set of automorphisms of G

that are conjugate to rϕsin Comm pGq, and we say that these automorphisms are

commensurably conjugate to ϕ. Any commensurated-conjugacy class is a union of

Aut pGq-conjugacy classes.

Questions. When are the commensurated-conjugacy classes strictly larger than Aut pGq-

conjugacy classes? When do they consist of ﬁnitely many Aut pGq-conjugacy classes?

We call the number of Aut pGq-conjugacy classes in the commensurated-conjugacy

class of ϕ, the class number of ϕ. This terminology is suggested and supported by

our familiar example, as we explain now.

Let G“Zn, so that Aut pGq “ GLnpZq, and Comm pGq “ GLnpQq. Choose

ϕPGLnpZqsuch that its characteristic polynomial χPZrXsis irreducible over

Z. All ψPGLnpZqwith the same characteristic polynomial as ϕare conjugate to

ϕin GLnpCq, hence in GLnpQqas well. Thus, they are commensurably conjugate.

The Latimer-MacDuﬀee theorem [17,28] states that the GLnpZq-conjugacy classes

of such elements are in correspondence with the ideal classes of the ring ZrXs{pχq.

Their cardinality, which is the order of the ideal class monoid of the ring, is called

the class number of the ring. In other words, the class number of ϕin our sense, is

the class number of the ring ZrXs{pχq. It is ﬁnite for all ϕPGLnpZq, larger than

the class number of the ring of integers OQrXs{pχq, and it is greater than 2 if and only

if the ring is not a principal ideal domain.

It is worth mentioning that ﬁniteness of class numbers is not true in general. A

countable group in which the class number of an automorphism is not ﬁnite is the

inﬁnite direct product of copies of Z{3: the automorphisms that are identity on

ﬁnitely many copies, and ﬂips on all the other copies, are all commensurably equal,

but not conjugate.

We turn our attention to non-abelian counterparts of the groups Zn. There are

two celebrated and important such classes: ﬁnite rank non-abelian free groups, and

fundamental groups of surfaces of higher genus. In this work we will consider the

latter.

Consider a closed orientable surface Σ of genus gě2. The extended mapping

class group of Σ is the homeomorphism group of Σ quotiented by its component of

identity HomeopΣq{Homeo0pΣq. By the Dehn-Nielsen-Baer theorem [11, Chapter

8], it is isomorphic to the outer automorphism group of π1pΣq. However in order to

make sense of commensuration, one must work at the level of automorphisms rather

than outer automorphisms – in particular one needs a base point pin Σ. We have:

Aut pπ1pΣ, pqq ↠Out pπ1pΣqq „

ÝÑ MCG˚pΣq

CLASS NUMBER FOR PSEUDO-ANOSOVS 3

The Nielsen-Thurston classiﬁcation of mapping classes of Σ distinguishes ﬁnite or-

der mapping classes, reducible mapping classes, and pseudo-Anosov mapping classes.

It is appropriate to regard pseudo-Anosov mapping classes as the correct placeholders

for truly irreducible mapping classes.

Deﬁnition 1.1. We will say that an automorphism is a pseudo-Anosov if its image

in the mapping class group under the above surjective homomorphism is a pseudo-

Anosov mapping class.

We establish the following uniform ﬁniteness for class numbers of pseudo-Anosov

automorphisms.

Theorem 1.2. If ϕis a pseudo-Anosov automorphism of the fundamental group of

a closed orientable surface of genus gě2, its class number is ﬁnite, bounded above

by pp168pg´1qq!q2g.

We think that it is striking that our bound does not depend on ϕ. For comparison,

in the case of Aut pZ2q » GL2pZq, by [17,28], one encounters class numbers of certain

rings of integers of real quadratic ﬁelds, which are conjectured to have interesting (but

elusive) behavior: Gauss’ class number problem conjectures that inﬁnitely many such

class numbers are 1, but these numbers are unbounded when the ﬁeld extension varies

among real quadratic extensions, as proved by Montgomery and Weinberger [22] and

quantitatively suggested by Cohen and Lenstra’s heuristics [9], and its revision by

Bartel and Lenstra [3]. We explain in the appendix that the class numbers of elements

of GL2pZqare indeed ﬁnite but unbounded, an argument explained to us by Sara

Checcoli.

Finiteness of the class number of a pseudo-Anosov is not a surprise. Here is a sim-

ple argument. Given ϕa pseudo-Anosov automorphism on a surface Σ, the stretch

factor, or entropy, lim

nÑ8 log |ϕnpγq|{ndoes not depend on γ‰1 in the fundamen-

tal group, nor on the metric up to quasi-isometry. Hence it is an invariant of the

commensurated-conjugacy class of ϕ. It also equals the translation length of ϕin the

Teichm¨uller space of the associated surface, and there are ﬁnitely many conjugacy

classes of a pseudo-Anosov that can have such translation length. (See [10] for

much ﬁner results on counting closed geodesics in moduli space.) Hence there can

only be ﬁnitely many conjugacy classes of ϕin the commensurated-conjugacy class

of ϕ. This argument does not, however, provide any uniform bound.

Our upper bound is very likely non-optimal. It seems that an optimal bound would

be given in terms of the value of a subgroup-growth function of certain orbifolds.

These are diﬃcult to estimate sharply, see [19].

Examples of commensurably conjugate automorphisms that are not conjugate are

not immediate. Our result and methods however suggest a ‘recipe’ for producing

examples of non-conjugate pseudo-Anosov automorphisms that are commensurably

4 FRANC¸ OIS DAHMANI AND MAHAN MJ

conjugate. We look at the induced action on the abelianization: elements that are

already conjugate in Aut pGqwould necessarily be conjugate in Aut pG{rG, Gsq, and

in the latter group, linear algebra tools may be applied to disprove the existence of

such a conjugacy.

Theorem 1.3. Let Σbe a closed orientable surface of genus 3, with a base point

p, and G“π1pΣ, pq. There exist ϕand ψpseudo-Anosov automorphisms of Gthat

are commensurably conjugate but whose images in Aut pG{rG, Gsq are not commen-

surably conjugate. In particular ϕand ψare not conjugate in Aut pGq.

Gauss famously conjectures (in his ‘class number problem’) that inﬁnitely many

real quadratic extensions of Qhave class number equal to one [26]. We observe that

in our non-abelian analogue, the corresponding question can be settled.

Theorem 1.4. There are inﬁnitely many commensurated classes of automorphisms

of hyperbolic 2-orbifolds, that have class number one.

Summary of the paper. After introducing the mapping tori involved and their

relation to commensurated conjugacy (in Section 2), we discuss in Section 3 their

representations as lattices in PSL2pCq. We show that, for diﬀerent commensurably

conjugate automorphisms, the groups of the mapping tori have representations that

commensurate the image of the surface group in PSL2pCq. We then use a result of

Leininger, Long and Reid [18] on the discreteness of the commensurator in PSL2pCq

of such an image, to obtain that all such representations land in a single lattice, and

we obtain Proposition 3.6. After ﬁnishing this work, we learned of the related, but

more restrictive notion of ﬁbered commensurability investigated by Calegari, Sun

and Wang [7, Section 2]. We then prove our main theorem at the end of Section

3. In Section 4 we provide an explicit example, and a recipe for general examples.

In Section 5 we address the class number one situation, proving Theorem 1.4. We

thought it worthwhile to attach an appendix, in which the algebraic number theory

relevant to the situation of Znis brieﬂy presented, in a way that allows for an analogy

with geometry and the case of surface group automorphisms. We brieﬂy survey orders

and ideal classes, provide an argument, explained to us by Sara Checcoli, as to why

the class numbers of matrices in GL2pZqare unbounded. We also explain a variant of

Latimer-MacDuﬀee’s proof that allows for a geometric perspective on the situation,

and is suited to an analogy with surface group automorphisms. Finally, we explain

this analogy and its limitations.

Acknowledgments. The authors thank the referee for valuable comments, and sug-

gestions. The ﬁrst author would like to thank Edgar Bering and Sara Checcoli for

discussions.

CLASS NUMBER FOR PSEUDO-ANOSOVS 5

2. Fundamental groups of mapping tori

The aim of this section is to prove Lemma 2.3 which converts the problem of

understanding commensurated-conjugacy classes of automorphisms of a group Gto

commensurability of the associated semi-direct product by Z. The one extra criterion

that is vital in Lemma 2.3 is that the commensuration of the associated semi-direct

product also commensurates G.

Lemma 2.1. If ϕ, ψ are automorphisms of Gin the same commensurated-conjugacy

class, there exists H, H1of ﬁnite index in G, and an isomorphism α:HÑH1such

that, ψpHq “ H, ϕpH1q “ H1,ψ|H“α´1˝ϕ˝α|H, where |Hdenotes restriction to

Proof. Let rαs P Comm pGqsuch that in Comm pGq,rψs“rαs´1rϕs rαs. Realize rαs

by an isomorphism α:T1ÑT2for T1, T2ﬁnite index subgroups of G.

Since rψs“rαs´1rϕs rαs, there is a further ﬁnite index subgroup Y1of T1on which

α´1˝ϕ˝α“ψ. Let Y2“αpY1q. Observe that ψpY1qmust be in T1but is perhaps

not Y1.

Let Hdenote the intersection of all subgroups in the Aut pGq-orbit of Y1, and H1

its image under α. Then H, H1are also of ﬁnite index. By construction, ψpreserves

H(as well as every automorphism). We still have α´1˝ϕ˝α“ψafter to restriction

H. It follows that ϕpH1q “ H1.

□

Given ϕPAut pGq, set xtϕyto be an abstract inﬁnite cyclic group, and consider

Γϕ“G¸ϕxtϕy. In this semi-direct product, we say that Gis the ﬁber.

Remark 2.2.This terminology is inspired by the motivating example in this paper,

where G“π1pSqwith Sa closed surface. Suppose that Φ : SÑSis a homeo-

morphism, so that ϕequals the automorphism of Ginduced by Φ after choosing a

base-point appropriately. Let Mdenote the 3-manifold given by the mapping torus

of Φ. Then Γϕis the fundamental group of M. Thus, from a topological point of

view, the relevant object to be considered is the mapping torus M. But for the

purposes of this section, it suﬃces to consider only the fundamental group of M.

Lemma 2.3. If ϕ, ψ are automorphisms of Gin the same commensurated-conjugacy

class, then, Γϕand Γψare commensurable by a homomorphism commensurating the

ﬁber.

Proof. Let H, H1be the subgroups obtained in the previous Lemma. We make semi-

direct products H¸ψxsyand H1¸ϕxqy. These two groups are isomorphic by an

isomorphism restricting to αon Hand sending sto q. Moreover these two groups

embed as ﬁnite index subgroups of Γϕand Γψsending sand qrespectively to tψand

tϕ.□

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CLASSNUMBERFORPSEUDO-ANOSOVSFRANC¸OISDAHMANIANDMAHANMJAbstract.GiventwoautomorphismsofagroupG,oneisinterestedinknowingwhethertheyareconjugateintheautomorphismgroupofG,orintheabstractcommensuratorofG,andhowthesetwopropertiesmaydiffer.WhenGisthefundamentalgroupofaclosedorientablesurface,wepresentaunif...

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