COMPATIBLE PANTS DECOMPOSITION FOR SL2C REPRESENTATIONS OF SURFACE GROUPS RENAUD DETCHERRY THOMAS LE FILS AND RAMANUJAN SANTHAROUBANE

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COMPATIBLE PANTS DECOMPOSITION FOR SL2(C)

REPRESENTATIONS OF SURFACE GROUPS

RENAUD DETCHERRY, THOMAS LE FILS AND RAMANUJAN SANTHAROUBANE

Abstract. For any irreducible representation of a surface group into SL2(C),

we show that there exists a pants decomposition where the restriction to any

pair of pants is irreducible and where no curve of the decomposition is sent to

a trace ±2 element. We prove a similar property for SO3-representations. We

also investigate the type of pants decomposition that can occur in this setting

for a given representation. This result was announced in [DS22], motivated by

the study of the Azumaya locus of the skein algebra of surfaces at roots of unity.

1. Introduction

Let Σ be a compact connected oriented surface (without boundary) of genus at

least two. Let ρ:π1(Σ) −→ G= SL2(C) or SO3be group homomorphism, we

start with the following deﬁnition :

Deﬁnition 1.1. A pants decomposition Pof Σ is called compatible with ρif for

any curve c∈ P,the elements ±ρ(c) are not unipotent and for any pants Pin P,

the restriction ρ|π1(P)is irreducible.

The purpose of this paper is to prove that any irreducible representation of

π1(Σ) into SL2(C) or SO3admits a compatible pants decomposition. Remark

that in the case of a SL2(C) representation, the compatibility condition can be

translated into a condition on the traces of the curves of the pants decomposition.

The ﬁrst condition is equivalent to Tr(ρ(c)) 6=±2,and the second is x2+y2+

z2−xyz −46= 0,where x, y, z are the traces of boundary curves of a pair of pants

in the decomposition. One type of pants decomposition important for us is the

sausage type which is a pants decomposition in the same orbit, under the action

of the mapping class group of Σ, as the one shown in Figure 1.

Theorem 1.2. Let ρ:π1(Σ) −→ G= SL2(C)or SO3be an irreducible represen-

tation. Then there is a compatible pants decomposition of Σfor ρ. Moreover, if

ρis a representation whose image is not conjugated to the quaternion Q8⊂SU2,

then there is a compatible pants decomposition for ρof sausage type.

The second part of this theorem was announced in [DS22, Theorem 1.5] without

a proof. The sausage type pants decomposition is very important in [DS22] and

Theorem 1.2 is key result to understand the Azumaya locus of the skein algebra of

Σ at roots of unity. Notice that the existence of compatible pants decomposition

arXiv:2210.09854v1 [math.GT] 18 Oct 2022

2 RENAUD DETCHERRY, THOMAS LE FILS AND RAMANUJAN SANTHAROUBANE

Figure 1. A sausage type pants decomposition of Σ

for non elementary representations in SL2(C) is a key step used by Gallo-Kapovich-

Marden in [GKM00] to prove that holonomies of CP1-structures are Zariski dense

in the SL2(C)-character variety of a given surface. It is quite intriguing that exact

same condition appeared in [DS22] in the context of quantum topology.

A result of Baba [Bab10] shows that the pants decompositions compatible with

a non-elementary representation ρarising from [GKM00] enables us to construct

explicitly all the projective structures with holonomy ρ. Our result for representa-

tions in SO3might also be used to describe the branched spherical structures with

given holonomy.

The question of ﬁnding compatible pants decomposition also emerged in the ﬁrst

named author’s thesis, motivated by Witten’s asymptotic expansion conjecture.

This conjecture expresses the asymptotics of WRT invariants of a 3-manifold M

as a sum of contributions associated to SU2representations of π1(M) and involving

Chern-Simons invariants and Reidemeister torsions. The ﬁrst author conjectures

that the geometric quantization techniques from [Det18] may be used to estimate

the contribution of representations that admit a compatible pants decomposition.

The proof of Theorem 1.2 is split in several steps. In Section 2 we deal with

representations which are non-elementary, representations with dense images in

SU2and representations with images in a non-compact dihedral group. For the

non elementary case, the existence of compatible pants decomposition is already

proved in [GKM00], we adapt these techniques to get the sausage type pants de-

composition. For the case of a representation with dense image in SU2, Theorem

1.2 is direct application of Previte-Xia’s result (see [PX02]) that proved that any

such representation has a dense orbit in the SU2character variety under the action

of the mapping class group. The last case is dealt by hand. In section 3, we treat

the remaining cases, namely representations with ﬁnite images. Such representa-

tions are classiﬁed, the proof is done by studying the orbits under the mapping

class group and building explicit compatible pants decomposition for each orbit.

Acknowledgements: Over the course of this work, the ﬁrst named author was

supported by the project “AlMaRe” (ANR-19-CE40-0001-01). The authors thank

Maxime Wolﬀ for very helpful conversations.

COMPATIBLE PANTS DECOMPOSITION FOR SURFACE GROUPS REPRESENTATIONS 3

2. Reduction to representations with finite image

In this section we reduce the proof of Theorem 1.2 to the case of representa-

tions ρwith ﬁnite images. More precisely we show Theorem 1.2 assuming that

Proposition 3.1 holds. This proposition will be proven in Section 3.

2.1. Mapping class group action. Let us begin with some observations on the

mapping class group action on sets of representations that will be used throughout

the rest of the paper. Recall that the mapping class group of Σ is deﬁned by

Mod(Σ) = Homeo+(Σ)/Homeo+

0(Σ).

The theorem of Dehn, Nielsen and Baer, see for example [FM12, Chapter 8],

states that its natural action on the fundamental group induces a group isomor-

phism Mod(Σ) →Out+(Σ) on the index two subgroup Out+(Σ) of Out(Σ) =

Aut(Σ)/Inn(Σ) induced by the automorphisms preserving orientation. Therefore

Mod(Σ) acts by precomposition as Out+(π1(Σ)) on the space of conjugacy classes

of representations Hom(π1(Σ), G)/G, for any group G.

A key observation to prove Theorem 1.2 is that this action preserves the set of

representations admitting a compatible pants decomposition. If ρ:π1(Σ) →Gis

a representation, we denote by [ρ] its conjugacy class.

Lemma 2.1. Suppose that ρ∞admits a compatible pants decomposition P. If

the conjugacy class [ρ∞]of ρ∞is in the closure of Mod(Σ) ·[ρ], then ρadmits a

compatible pants decomposition of the type of P.

Proof. For a given pants decomposition P of Σ, let us denote by C(P) the set of

conjugacy classes of representations π1(Σ) →Gcompatible with P. It follows from

the deﬁnition of compatibility that these sets are open. Therefore there exists a

representation in (Mod(Σ) ·[ρ]) ∩ C(P). Hence there exists f∈Mod(Σ) such that

f·[ρ]∈ C(P) and therefore [ρ]∈ C(f−1· P). 

We will thus prove Theorem 1.2 by studying the orbits of representations π1(Σ) →

SL2(C). We will use diﬀerent methods depending on the image of the representa-

tion we wish to study.

2.2. Non-elementary case. Let us begin with the case where ρis a non-elementary

representation, i.e. the action of its image on the Riemann sphere CP1by M¨obius

transformations has no ﬁnite orbit. We can in that case adapt the strategy of

Gallo, Kapovich and Marden in [GKM00, Part A] to ﬁnd a compatible pants

decomposition.

Proposition 2.2. Let ρ:π1(Σ) −→ PSL2(C)be a non-elementary representation.

For any trivalent graph Γwith 3g−3edges that has at least one one-edge loop,

there is a pants decomposition of Σ,with associated graph Γwhich is compatible

with ρ.

4 RENAUD DETCHERRY, THOMAS LE FILS AND RAMANUJAN SANTHAROUBANE

Proof. Let us begin by recalling the main steps of the construction by Gallo,

Kapovich and Marden in [GKM00] of a Schottky pants decomposition for ρ: a

pants decomposition such that the restriction of ρto each pair of pants is an iso-

morphism onto a Schottky group. Note that this construction of Gallo, Kapovich

and Marden works for every non-elementary representation π1(Σ) −→ PSL2(C)

except in genus g= 2 for the pentagon representations. We suppose for now that

ρis not a pentagon representation.

The ﬁrst step is to ﬁnd special handle in Σ. That is a handle Hwhose fun-

damental group π1(H) is sent by ρonto a non-elementary subgroup of SL2(C).

This handle Hallows us to ﬁnd g−1 disjoint simple curves away from it, that are

sent by ρto loxodromic elements. Cutting the surface along those curves leads

to a genus one surface with 2g−2 boundary components. Choosing any two of

these components, we ﬁnd a curve separating them from rest of the surface, and

such that the restriction of ρto the pair of pants they bound is an isomorphism

onto a Schottky group. Cutting along this curve takes oﬀ a pair of pants and

gives a genus one surface with 2g−3 boundary components. We repeat the same

procedure until we get a genus one surface with two boundary components where

a special cutting process is applied to get a Schottky pants decomposition. This

process ﬁnds a curve bounding the two boundary components and cuts the handle.

We now show that choosing wisely the curves at each step allows us to create a

decomposition with any trivalent graph with 3g−3 edges and at least one one-edge

loop. We thus are reduced to the following combinatorial lemma.

Lemma 2.3. Any trivalent graph Γwith 3g−3edges that has at least one one-edge

loop can be created by this procedure.

Proof. We start from a genus one surface Σ with 2g−2 boundary components that

come from cutting g−1 curves from a closed surface. Let us label the boundary

components by integers 1 6k6g−1 with the same label if they come from

cutting the same curve. Pick a one-edge loop e. Let us consider the graph Γ0that

is the graph Γ with eand all the edges connected to it removed, as for example

in Figure 2. Let us cut Γ0along g−1 edges that do not disconnect it. We obtain

ΓΓ0

Figure 2. Example of a trivalent graph Γ and Γ0.

a new graph Γ0with 2g−2 boundary components that we label with integers

16k6g−1. We require that two boundaries have the same label if they come

from the same edge, see the left side of Figure 3.

The graph Γ0has 2g−2 boundary components and 2g−3 vertices. Therefore

one of the vertices has two boundary components, let us choose such a vertex f.

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COMPATIBLEPANTSDECOMPOSITIONFORSL2(C)REPRESENTATIONSOFSURFACEGROUPSRENAUDDETCHERRY,THOMASLEFILSANDRAMANUJANSANTHAROUBANEAbstract.ForanyirreduciblerepresentationofasurfacegroupintoSL2(C),weshowthatthereexistsapantsdecompositionwheretherestrictiontoanypairofpantsisirreducibleandwherenocurveofthedecomp...

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COMPATIBLE PANTS DECOMPOSITION FOR SL2C REPRESENTATIONS OF SURFACE GROUPS RENAUD DETCHERRY THOMAS LE FILS AND RAMANUJAN SANTHAROUBANE

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