Asymptotic of Twisted Alexander Polynomials and Hyperbolic Volume

Home , Hyperbolic volume

Asymptotic of twisted polynomials

L. Benard, J. Dubois, M. Heusener, J. Porti Asymptotic of twisted Alexander polynomials and hyperbolic volume

L. Benard, J. Dubois, M. Heusener, J. Porti

April 23, 2020 Hyperbolic: 3 M ' H /Γ Where the holonomy map

∼ ρ: π1(M) −→ Γ

identiﬁes π1(M) with a discrete subgroup

Γ ⊂ (P) SL2(C)

3 which acts on H by isometries.

Introduction

Asymptotic of twisted polynomials

L. Benard, J. We consider M a hyperbolic 3-manifold of ﬁnite volume. Dubois, M. Heusener, J. Porti Where the holonomy map

∼ ρ: π1(M) −→ Γ

identiﬁes π1(M) with a discrete subgroup

Γ ⊂ (P) SL2(C)

3 which acts on H by isometries.

Introduction

Asymptotic of twisted polynomials

L. Benard, J. We consider M a hyperbolic 3-manifold of ﬁnite volume. Dubois, M. Heusener, J. Hyperbolic: Porti 3 M ' H /Γ 3 which acts on H by isometries.

Introduction

Asymptotic of twisted polynomials

L. Benard, J. We consider M a hyperbolic 3-manifold of ﬁnite volume. Dubois, M. Heusener, J. Hyperbolic: Porti 3 M ' H /Γ Where the holonomy map

∼ ρ: π1(M) −→ Γ

identiﬁes π1(M) with a discrete subgroup

Γ ⊂ (P) SL2(C) Introduction

Asymptotic of twisted polynomials

L. Benard, J. We consider M a hyperbolic 3-manifold of ﬁnite volume. Dubois, M. Heusener, J. Hyperbolic: Porti 3 M ' H /Γ Where the holonomy map

∼ ρ: π1(M) −→ Γ

identiﬁes π1(M) with a discrete subgroup

Γ ⊂ (P) SL2(C)

3 which acts on H by isometries. It follows from Mostow rigidity that the volume of M is a topological invariant.

Hyperbolic manifolds

Asymptotic of twisted We consider M a hyperbolic 3-manifold of ﬁnite volume, for polynomials instance L. Benard, J. Dubois, M. Heusener, J. Example (The ﬁgure-eight knot) Porti Hyperbolic manifolds

Asymptotic of twisted We consider M a hyperbolic 3-manifold of ﬁnite volume, for polynomials instance L. Benard, J. Dubois, M. Heusener, J. Example (The ﬁgure-eight knot) Porti

It follows from Mostow rigidity that the volume of M is a topological invariant. where M is a hyperbolic manifold, and ρ: π1(M) → SL2(C) its holonomy. It reﬁnes the cohomological information contained in some cellular complex

d d C 0(M, ρ) −→0 C 1(M, ρ) −→1 C 2(M, ρ) → ...

It should be thought as the alternating product of the ”determinants” of the boundary operators di :

Y (−1)i ∗ tor(M, ρ) = det(di ) ∈ C i

Reidemeister torsion

Asymptotic of twisted An other topological invariant is the Reidemeister torsion of the polynomials pair L. Benard, J. Dubois, M. (M, ρ) Heusener, J. Porti It reﬁnes the cohomological information contained in some cellular complex

d d C 0(M, ρ) −→0 C 1(M, ρ) −→1 C 2(M, ρ) → ...

It should be thought as the alternating product of the ”determinants” of the boundary operators di :

Y (−1)i ∗ tor(M, ρ) = det(di ) ∈ C i

Reidemeister torsion

Asymptotic of twisted An other topological invariant is the Reidemeister torsion of the polynomials pair L. Benard, J. Dubois, M. (M, ρ) Heusener, J. Porti where M is a hyperbolic manifold, and ρ: π1(M) → SL2(C) its holonomy. It should be thought as the alternating product of the ”determinants” of the boundary operators di :

Y (−1)i ∗ tor(M, ρ) = det(di ) ∈ C i

Reidemeister torsion

Asymptotic of twisted An other topological invariant is the Reidemeister torsion of the polynomials pair L. Benard, J. Dubois, M. (M, ρ) Heusener, J. Porti where M is a hyperbolic manifold, and ρ: π1(M) → SL2(C) its holonomy. It reﬁnes the cohomological information contained in some cellular complex

d d C 0(M, ρ) −→0 C 1(M, ρ) −→1 C 2(M, ρ) → ... Reidemeister torsion

Asymptotic of twisted An other topological invariant is the Reidemeister torsion of the polynomials pair L. Benard, J. Dubois, M. (M, ρ) Heusener, J. Porti where M is a hyperbolic manifold, and ρ: π1(M) → SL2(C) its holonomy. It reﬁnes the cohomological information contained in some cellular complex

d d C 0(M, ρ) −→0 C 1(M, ρ) −→1 C 2(M, ρ) → ...

It should be thought as the alternating product of the ”determinants” of the boundary operators di :

Y (−1)i ∗ tor(M, ρ) = det(di ) ∈ C i We introduce some notation: the (n − 1)-th symmetric power Symn−1 denotes the unique irreducible embedding

Symn−1 : SL2(C) ,→ SLn(C) induced by the isomorphism 2 n Symn−1(C ) ' C For instance  λn−1  n−3 λ 0 λ Sym =  ..  n−1 0 λ−1  .  λ3−n λ1−n

Torsion and volume

Asymptotic of The ﬁrst natural (informal) question is the following: twisted polynomials Question L. Benard, J. Dubois, M. Heusener, J. Is there a relation between tor(M, ρ) and Vol(M)? Porti For instance  λn−1  n−3 λ 0 λ Sym =  ..  n−1 0 λ−1  .  λ3−n λ1−n

Torsion and volume

Asymptotic of The ﬁrst natural (informal) question is the following: twisted polynomials Question L. Benard, J. Dubois, M. Heusener, J. Is there a relation between tor(M, ρ) and Vol(M)? Porti We introduce some notation: the (n − 1)-th symmetric power Symn−1 denotes the unique irreducible embedding

Symn−1 : SL2(C) ,→ SLn(C) induced by the isomorphism 2 n Symn−1(C ) ' C Torsion and volume

Asymptotic of The ﬁrst natural (informal) question is the following: twisted polynomials Question L. Benard, J. Dubois, M. Heusener, J. Is there a relation between tor(M, ρ) and Vol(M)? Porti We introduce some notation: the (n − 1)-th symmetric power Symn−1 denotes the unique irreducible embedding

Symn−1 : SL2(C) ,→ SLn(C) induced by the isomorphism 2 n Symn−1(C ) ' C For instance  λn−1  n−3 λ 0 λ Sym =  ..  n−1 0 λ−1  .  λ3−n λ1−n Asymptotic of torsions

Asymptotic of twisted polynomials Previous question has been answered positively as follows: L. Benard, J. Dubois, M. Theorem (M¨uller’12 for the compact case, Heusener, J. Porti Menal-Ferrer–Porti ’14 for the general case) Denote by

ρ Symn−1 ρn : π1(M) −→ SL2(C) −−−−→ SLn(C)

the (n − 1) symmetric power of the holonomy representation of a hyperbolic manifold M. The following holds:

log | tor(M, ρ )| Vol(M) lim n = n→∞ n2 4π First recall that the holonomy representation ρ is part of a moduli space of deformations of geometric structures, the deformation variety, or character variety. In a few words, despite the holonomy representation corresponds to the unique (thanks to Mostow rigidity) complete hyperbolic structure on M, one can deform this structure into non-complete ones, yielding a moduli space. Moreover, this character variety is an analytic (even algebraic) variety, equipped with analytic functions

tor: [%] 7→ tor(M,%) Vol: [%] 7→ Vol(%)

with Vol(ρ) = Vol(M) when ρ is the holonomy.

Our motivations

Asymptotic of twisted The ﬁrst question it raises, which is our original motivation, is polynomials the following. L. Benard, J. Dubois, M. Heusener, J. Porti In a few words, despite the holonomy representation corresponds to the unique (thanks to Mostow rigidity) complete hyperbolic structure on M, one can deform this structure into non-complete ones, yielding a moduli space. Moreover, this character variety is an analytic (even algebraic) variety, equipped with analytic functions

tor: [%] 7→ tor(M,%) Vol: [%] 7→ Vol(%)

with Vol(ρ) = Vol(M) when ρ is the holonomy.

Our motivations

tor: [%] 7→ tor(M,%) Vol: [%] 7→ Vol(%)

with Vol(ρ) = Vol(M) when ρ is the holonomy.

Our motivations

Asymptotic of twisted The ﬁrst question it raises, which is our original motivation, is polynomials the following. L. Benard, J. Dubois, M. First recall that the holonomy representation ρ is part of a Heusener, J. Porti moduli space of deformations of geometric structures, the deformation variety, or character variety. In a few words, despite the holonomy representation corresponds to the unique (thanks to Mostow rigidity) complete hyperbolic structure on M, one can deform this structure into non-complete ones, yielding a moduli space. Our motivations

tor: [%] 7→ tor(M,%) Vol: [%] 7→ Vol(%)

with Vol(ρ) = Vol(M) when ρ is the holonomy. It turns out that most of the techniques of their proofs fall down when % is not the holonomy representation.

Deformation

Asymptotic of twisted polynomials A natural question is then: L. Benard, J. Dubois, M. Heusener, J. Question Porti Can we ”deform” the statement of M¨uller and Menal-Ferrer–Porti into: log | tor(M,% )| Vol(%) lim n = n→∞ n2 4π

for any %: π1(M) → SL2(C) close to the holonomy representation ρ? Deformation

for any %: π1(M) → SL2(C) close to the holonomy representation ρ?

It turns out that most of the techniques of their proofs fall down when % is not the holonomy representation. Remark

In the sequel we will always assume that b1(M) ≥ 1.

For sake of simplicity, assume that b1(M) = 1. Let m ∈ π1(M) such that [m] is a generator of H1(M)/(Tor H1(M)) ' Z. 1 1 Given ζ in the unit circle S ; we denote by χζ : π1(M) → S the homomorphism that sends m to ζ. It induces a new family of representations

1 ρn ⊗ χζ : π1(M) → SLn(C) ⊗ S γ 7→ ρn(γ)χζ (γ)

We will consider the torsions of the twisted representations tor(M, ρn ⊗ ζ).

A more aﬀordable ﬁrst step

Asymptotic of To begin with, we consider the following more simple family of twisted polynomials deformations. L. Benard, J. Dubois, M. Heusener, J. Porti For sake of simplicity, assume that b1(M) = 1. Let m ∈ π1(M) such that [m] is a generator of H1(M)/(Tor H1(M)) ' Z. 1 1 Given ζ in the unit circle S ; we denote by χζ : π1(M) → S the homomorphism that sends m to ζ. It induces a new family of representations

1 ρn ⊗ χζ : π1(M) → SLn(C) ⊗ S γ 7→ ρn(γ)χζ (γ)

We will consider the torsions of the twisted representations tor(M, ρn ⊗ ζ).

A more aﬀordable ﬁrst step

Asymptotic of To begin with, we consider the following more simple family of twisted polynomials deformations. L. Benard, J. Dubois, M. Remark Heusener, J. Porti In the sequel we will always assume that b1(M) ≥ 1. 1 1 Given ζ in the unit circle S ; we denote by χζ : π1(M) → S the homomorphism that sends m to ζ. It induces a new family of representations

1 ρn ⊗ χζ : π1(M) → SLn(C) ⊗ S γ 7→ ρn(γ)χζ (γ)

We will consider the torsions of the twisted representations tor(M, ρn ⊗ ζ).

A more aﬀordable ﬁrst step

For sake of simplicity, assume that b1(M) = 1. Let m ∈ π1(M) such that [m] is a generator of H1(M)/(Tor H1(M)) ' Z. We will consider the torsions of the twisted representations tor(M, ρn ⊗ ζ).

A more aﬀordable ﬁrst step

1 ρn ⊗ χζ : π1(M) → SLn(C) ⊗ S γ 7→ ρn(γ)χζ (γ) A more aﬀordable ﬁrst step

1 ρn ⊗ χζ : π1(M) → SLn(C) ⊗ S γ 7→ ρn(γ)χζ (γ)

We will consider the torsions of the twisted representations tor(M, ρn ⊗ ζ). These topological invariants have a (partially conjectural) strong detection power. Our ﬁrst result is the following: Theorem (BDHP ’19) 1 For any ζ on the unit circle S ,

n |∆M (ζ)| = | tor(M, ρn ⊗ χζ )|.

n In particular, the polynomials ∆M have no roots on the unit circle.

Twisted Alexander polynomials

Asymptotic of twisted n ±1 polynomials There is a family of polynomials ∆M (t) ∈ C[t ], the L. Benard, J. ρn-twisted Alexander polynomials, that reﬁne the construction Dubois, M. Heusener, J. of the Alexander polynomial for knots (Wada, Lin...) Porti Our ﬁrst result is the following: Theorem (BDHP ’19) 1 For any ζ on the unit circle S ,

n |∆M (ζ)| = | tor(M, ρn ⊗ χζ )|.

n In particular, the polynomials ∆M have no roots on the unit circle.

Twisted Alexander polynomials

Asymptotic of twisted n ±1 polynomials There is a family of polynomials ∆M (t) ∈ C[t ], the L. Benard, J. ρn-twisted Alexander polynomials, that reﬁne the construction Dubois, M. Heusener, J. of the Alexander polynomial for knots (Wada, Lin...) Porti These topological invariants have a (partially conjectural) strong detection power. Twisted Alexander polynomials

Asymptotic of twisted n ±1 polynomials There is a family of polynomials ∆M (t) ∈ C[t ], the L. Benard, J. ρn-twisted Alexander polynomials, that reﬁne the construction Dubois, M. Heusener, J. of the Alexander polynomial for knots (Wada, Lin...) Porti These topological invariants have a (partially conjectural) strong detection power. Our ﬁrst result is the following: Theorem (BDHP ’19) 1 For any ζ on the unit circle S ,

n |∆M (ζ)| = | tor(M, ρn ⊗ χζ )|.

n In particular, the polynomials ∆M have no roots on the unit circle. Taking the log of the modulus, t=1, k=2 log |∆n (1)| = log |∆n (1)| + log |∆n (−1)| M M M n 2 Replacing ∆M (1) by tor(M, ρn) and dividing by n log | tor(M, ρ )| log | tor(M, ρ )| + log |∆n (−1)| n = n M n2 n2 Taking the limit as n → ∞ and applying previous theorem: 2 Vol(M) Vol(M) log |∆n (−1)| = + lim M 4π 4π n→∞ n2

A suggestive computation

Asymptotic of Those polynomials satisfy a lot of nice symmetry properties. twisted polynomials Among them, if M → M is a cyclic k-sheeted covering map L. Benard, J. then for any t Dubois, M. Y Heusener, J. ∆n (t) = ∆n (ζt) Porti M M ζk =1 n 2 Replacing ∆M (1) by tor(M, ρn) and dividing by n log | tor(M, ρ )| log | tor(M, ρ )| + log |∆n (−1)| n = n M n2 n2 Taking the limit as n → ∞ and applying previous theorem: 2 Vol(M) Vol(M) log |∆n (−1)| = + lim M 4π 4π n→∞ n2

A suggestive computation

Asymptotic of Those polynomials satisfy a lot of nice symmetry properties. twisted polynomials Among them, if M → M is a cyclic k-sheeted covering map L. Benard, J. then for any t Dubois, M. Y Heusener, J. ∆n (t) = ∆n (ζt) Porti M M ζk =1 Taking the log of the modulus, t=1, k=2 log |∆n (1)| = log |∆n (1)| + log |∆n (−1)| M M M Taking the limit as n → ∞ and applying previous theorem: 2 Vol(M) Vol(M) log |∆n (−1)| = + lim M 4π 4π n→∞ n2

A suggestive computation

Asymptotic of Those polynomials satisfy a lot of nice symmetry properties. twisted polynomials Among them, if M → M is a cyclic k-sheeted covering map L. Benard, J. then for any t Dubois, M. Y Heusener, J. ∆n (t) = ∆n (ζt) Porti M M ζk =1 Taking the log of the modulus, t=1, k=2 log |∆n (1)| = log |∆n (1)| + log |∆n (−1)| M M M n 2 Replacing ∆M (1) by tor(M, ρn) and dividing by n log | tor(M, ρ )| log | tor(M, ρ )| + log |∆n (−1)| n = n M n2 n2 A suggestive computation

Asymptotic of Those polynomials satisfy a lot of nice symmetry properties. twisted polynomials Among them, if M → M is a cyclic k-sheeted covering map L. Benard, J. then for any t Dubois, M. Y Heusener, J. ∆n (t) = ∆n (ζt) Porti M M ζk =1 Taking the log of the modulus, t=1, k=2 log |∆n (1)| = log |∆n (1)| + log |∆n (−1)| M M M n 2 Replacing ∆M (1) by tor(M, ρn) and dividing by n log | tor(M, ρ )| log | tor(M, ρ )| + log |∆n (−1)| n = n M n2 n2 Taking the limit as n → ∞ and applying previous theorem: 2 Vol(M) Vol(M) log |∆n (−1)| = + lim M 4π 4π n→∞ n2 Our main result is: Theorem (BDHP ’19) For any ζ on the unit circle,

log |∆n (ζ)| Vol(M) lim M = n→∞ n2 4π uniformly in ζ.

Our main theorem

Asymptotic of twisted polynomials One deduces L. Benard, J. n Dubois, M. log |∆M (ζ)| Vol(M) Heusener, J. lim 2 = Porti n→∞ n 4π for ζ = −1. In fact, the same trick works as well for ζ root of order 3, 4, 6... but not more. Our main theorem

log |∆n (ζ)| Vol(M) lim M = n→∞ n2 4π uniformly in ζ. The key ingredient is the following theorem: Theorem (Cheeger ’77, M¨uller’78, ’91, Bismut–Zhang ’91)

Let N be a compact (3-)manifold, %: π1(N) → GLn(C) a unimodular representation (i. e. | det ρ(γ)| = 1 for any γ in π1(N)). Let T (M, E%) denote the analytic torsion of the ﬂat bundle E% associated to %. Then

| tor(M,%)| = T (M, E%).

Cheeger–M¨ullertheorem

Asymptotic of twisted polynomials As we said before, to prove this theorem we need to study the L. Benard, J. Dubois, M. asymptotic of the Reidemeister torsions tor(M, ρn ⊗ χζ ) as n Heusener, J. Porti goes to ∞. Cheeger–M¨ullertheorem

| tor(M,%)| = T (M, E%). The ﬂat vector bundle Eρn⊗χζ is the quotient 3 n 3 H ×ρn⊗χζ C of the trivial bundle of rank n on H by the equivalence relation

(xe, v) ∼ (γ · xe, ρn ⊗ χζ (γ)v)

An Eρn⊗χζ -valued function (or diﬀerential form)

f : M → Eρn⊗χζ is a π1(M)-equivariant function 3 n ∗ f : H → C . We denote by Ω (M, Eρn⊗χζ ) the complex

of Eρn⊗χζ -valued diﬀerential forms on M.

Idea of the proof: the analytic torsion.

Asymptotic of For the proof we assume that M is compact, so that we can twisted polynomials use the Cheeger–M¨ullertheorem: since ρn ⊗ χζ is unimodular, L. Benard, J. we are led to consider the sequence of analytic torsions Dubois, M. Heusener, J. Porti T (M, Eρn⊗χζ ) n. An Eρn⊗χζ -valued function (or diﬀerential form)

f : M → Eρn⊗χζ is a π1(M)-equivariant function 3 n ∗ f : H → C . We denote by Ω (M, Eρn⊗χζ ) the complex

of Eρn⊗χζ -valued diﬀerential forms on M.

Idea of the proof: the analytic torsion.

The ﬂat vector bundle Eρn⊗χζ is the quotient 3 n 3 H ×ρn⊗χζ C of the trivial bundle of rank n on H by the equivalence relation

(xe, v) ∼ (γ · xe, ρn ⊗ χζ (γ)v) Idea of the proof: the analytic torsion.

The ﬂat vector bundle Eρn⊗χζ is the quotient 3 n 3 H ×ρn⊗χζ C of the trivial bundle of rank n on H by the equivalence relation

(xe, v) ∼ (γ · xe, ρn ⊗ χζ (γ)v)

An Eρn⊗χζ -valued function (or diﬀerential form)

f : M → Eρn⊗χζ is a π1(M)-equivariant function 3 n ∗ f : H → C . We denote by Ω (M, Eρn⊗χζ ) the complex

of Eρn⊗χζ -valued differential forms on M. The Laplacian is the operator ∗ ∗ k k ∆k = dk−1 dk−1 + dk dk :Ω (M, Eρn⊗χζ ) → Ω (M, Eρn⊗χζ ). Definition The analytic torsion is defined as

3 Y k k (−1) 2 T (M, Eρn⊗χζ ) = (det ∆k ) . k=0

Idea of the proof: the analytic torsion 2.

Asymptotic of twisted polynomials

L. Benard, J. Dubois, M. 0 d0 1 d1 2 d2 Heusener, J. Ω (M, Eρn⊗χζ ) / Ω (M, Eρn⊗χζ ) / Ω (M, Eρn⊗χζ ) / ... Porti The Laplacian is the operator ∗ ∗ k k ∆k = dk−1 dk−1 + dk dk :Ω (M, Eρn⊗χζ ) → Ω (M, Eρn⊗χζ ). Deﬁnition The analytic torsion is deﬁned as

3 Y k k (−1) 2 T (M, Eρn⊗χζ ) = (det ∆k ) . k=0

Idea of the proof: the analytic torsion 2.

Asymptotic of twisted polynomials 0 d0 1 d1 2 d2 L. Benard, J. Ω (M, Eρn⊗χζ ) / Ω (M, Eρn⊗χζ ) / Ω (M, Eρn⊗χζ ) / ... Dubois, M. Heusener, J. Porti Once again, we wish to compute the (analytic) torsion as Y (−1)i T (M, Eρn⊗χζ ) = det(di ) i The Laplacian is the operator ∗ ∗ k k ∆k = dk−1 dk−1 + dk dk :Ω (M, Eρn⊗χζ ) → Ω (M, Eρn⊗χζ ). Deﬁnition The analytic torsion is deﬁned as

3 Y k k (−1) 2 T (M, Eρn⊗χζ ) = (det ∆k ) . k=0

Idea of the proof: the analytic torsion 2.

Asymptotic of twisted polynomials

L. Benard, J. 0 d0 1 d1 2 d2 Dubois, M. Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / ... ! < −2ex, 0ex > Heusener, J. n ζ n ζ n ζ Porti m ∗ m ∗ m ∗ d0 d1 d2 Deﬁnition The analytic torsion is deﬁned as

3 Y k k (−1) 2 T (M, Eρn⊗χζ ) = (det ∆k ) . k=0

Idea of the proof: the analytic torsion 2.

Asymptotic of twisted polynomials

L. Benard, J. 0 d0 1 d1 2 d2 Dubois, M. Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / ... ! < −2ex, 0ex > Heusener, J. n ζ n ζ n ζ Porti m ∗ m ∗ m ∗ d0 d1 d2

The Laplacian is the operator ∗ ∗ k k ∆k = dk−1 dk−1 + dk dk :Ω (M, Eρn⊗χζ ) → Ω (M, Eρn⊗χζ ). Idea of the proof: the analytic torsion 2.

Asymptotic of twisted polynomials

L. Benard, J. 0 d0 1 d1 2 d2 Dubois, M. Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / Ω (M, Eρ ⊗χ ) / ... ! < −2ex, 0ex > Heusener, J. n ζ n ζ n ζ Porti m ∗ m ∗ m ∗ d0 d1 d2

The Laplacian is the operator ∗ ∗ k k ∆k = dk−1 dk−1 + dk dk :Ω (M, Eρn⊗χζ ) → Ω (M, Eρn⊗χζ ). Deﬁnition The analytic torsion is deﬁned as

3 Y k k (−1) 2 T (M, Eρn⊗χζ ) = (det ∆k ) . k=0 P −s First consider the zeta regularisation: the series λm converges for the real part of s big enough, and its derivative P at s = 0 formally equals − log λm. Now use the Mellin transform: Z ∞ e−tλts−1dt = λ−s Γ(s) 0 with Γ(s) the gamma-function (meromorphic, with simple pole at 0). The candidate for log det ∆ is

R ∞ P −tλm s−1 ! d m e t dt − 0 ds Γ(s) s=0

Determinant and trace

Asymptotic of But what is the meaning of det ∆ ? The spectrum of the twisted polynomials Laplacian consists of eigenvalues L. Benard, J. {0 < λ ≤ λ ≤ ... ≤ λ → +∞}. Formally, we write Dubois, M. 1 2 m Heusener, J. log det ∆ = Tr log ∆ = P log λ but the latter makes no sense. Porti m Now use the Mellin transform: Z ∞ e−tλts−1dt = λ−s Γ(s) 0 with Γ(s) the gamma-function (meromorphic, with simple pole at 0). The candidate for log det ∆ is

R ∞ P −tλm s−1 ! d m e t dt − 0 ds Γ(s) s=0

Determinant and trace

Asymptotic of But what is the meaning of det ∆ ? The spectrum of the twisted polynomials Laplacian consists of eigenvalues L. Benard, J. {0 < λ ≤ λ ≤ ... ≤ λ → +∞}. Formally, we write Dubois, M. 1 2 m Heusener, J. log det ∆ = Tr log ∆ = P log λ but the latter makes no sense. Porti m P −s First consider the zeta regularisation: the series λm converges for the real part of s big enough, and its derivative P at s = 0 formally equals − log λm. The candidate for log det ∆ is

R ∞ P −tλm s−1 ! d m e t dt − 0 ds Γ(s) s=0

Determinant and trace

Asymptotic of But what is the meaning of det ∆ ? The spectrum of the twisted polynomials Laplacian consists of eigenvalues L. Benard, J. {0 < λ ≤ λ ≤ ... ≤ λ → +∞}. Formally, we write Dubois, M. 1 2 m Heusener, J. log det ∆ = Tr log ∆ = P log λ but the latter makes no sense. Porti m P −s First consider the zeta regularisation: the series λm converges for the real part of s big enough, and its derivative P at s = 0 formally equals − log λm. Now use the Mellin transform: Z ∞ e−tλts−1dt = λ−s Γ(s) 0 with Γ(s) the gamma-function (meromorphic, with simple pole at 0). Determinant and trace

Asymptotic of But what is the meaning of det ∆ ? The spectrum of the twisted polynomials Laplacian consists of eigenvalues L. Benard, J. {0 < λ ≤ λ ≤ ... ≤ λ → +∞}. Formally, we write Dubois, M. 1 2 m Heusener, J. log det ∆ = Tr log ∆ = P log λ but the latter makes no sense. Porti m P −s First consider the zeta regularisation: the series λm converges for the real part of s big enough, and its derivative P at s = 0 formally equals − log λm. Now use the Mellin transform: Z ∞ e−tλts−1dt = λ−s Γ(s) 0 with Γ(s) the gamma-function (meromorphic, with simple pole at 0). The candidate for log det ∆ is

R ∞ P −tλm s−1 ! d m e t dt − 0 ds Γ(s) s=0 n Denoting (∆ )n for the family of Laplacians acting of forms valued in the family of bundles E , we need to study ρn⊗χζ n the asymptotic behavior of the heat traces of this family as n goes to inﬁnity. Since the Laplacians operators are equivariant under the action of π1(M), we can decompose the heat traces on translated of a 3 fundamental domain F ⊂ H for M: Z −t∆n X n Tr e = χζ (γ) ht (xe, γ · xe) F [γ]∈[π1(M)]

The heat operator

Asymptotic of twisted We have just seen that to make sense of the determinant of polynomials the Laplace operator det ∆, one needs to study the sum L. Benard, J. Dubois, M. P e−tλm , which is the trace of the heat operator e−t∆ (and Heusener, J. m Porti is well-deﬁned for any t > 0). n Denoting (∆ )n for the family of Laplacians acting of forms valued in the family of bundles E , we need to study ρn⊗χζ n the asymptotic behavior of the heat traces of this family as n goes to inﬁnity. Since the Laplacians operators are equivariant under the action of π1(M), we can decompose the heat traces on translated of a 3 fundamental domain F ⊂ H for M: Z −t∆n X n Tr e = χζ (γ) ht (xe, γ · xe) F [γ]∈[π1(M)] The last series is know as a Ruelle zeta function, and part of the computations goes through a proof of Fried theorem, which relates those Ruelle zeta functions with the analytic torsion. The end of the proof deals with the convergence of the above sum as n goes to ∞. The delicate points are uniformity of the convergence.

Ruelle zeta functions

Asymptotic of twisted polynomials After (many) more computations, one obtains the formula: L. Benard, J. Dubois, M. 2 n Heusener, J. n Vol(M) X X − kλ(γ) Porti log T (M, E ) = − log 1 − χ (γ)e 2 ρn⊗χζ 4π ζ k=1 [γ]6=1 (1) where λ(γ) is the complex length of γ i.e. ρ(γ) ∼ eλ(γ)/2 0 . 0 e−λ(γ)/2 The last series is know as a Ruelle zeta function, and part of the computations goes through a proof of Fried theorem, which relates those Ruelle zeta functions with the analytic torsion. Ruelle zeta functions

Asymptotic of twisted polynomials

L. Benard, J. Dubois, M. Heusener, J. Porti To go to the non-compact case, one approximates the manifold M by a sequence a compact manifolds Mp, and one needs uniformity in p in (1). Dividing by n2 and taking the limit ﬁnishes the proof.

Asymptotic of twisted polynomials

L. Benard, J. Dubois, M. Heusener, J. Porti To go to the non-compact case, one approximates the manifold M by a sequence a compact manifolds Mp, and one needs uniformity in p in (1). We also want uniformity in ζ. Dividing by n2 and taking the limit ﬁnishes the proof. The representation ρn restricts to a representation of π1(Σ), and the monodromy acts on ρn,Σ by conjugation. In other words, [ρn,Σ] is a ﬁxed point for the action of φ on the character variety of Σ. We prove the following: Theorem (BDHP’19)

The action of the monodromy φ on [ρn,Σ] has hyperbolic dynamic. Namely its tangent map has no eigenvalues of modulus one.

A dynamical application of our ﬁrst theorem

Asymptotic of twisted polynomials Assume that M is fibered: M = Σ × [0, 1]/(x, 0) ∼ (φ(x), 1) L. Benard, J. Dubois, M. for some surface Σ and some diffeomorphism φ:Σ → Σ, called Heusener, J. Porti the monodromy. In other words, [ρn,Σ] is a fixed point for the action of φ on the character variety of Σ. We prove the following: Theorem (BDHP’19)

The action of the monodromy φ on [ρn,Σ] has hyperbolic dynamic. Namely its tangent map has no eigenvalues of modulus one.

A dynamical application of our ﬁrst theorem

Asymptotic of twisted polynomials Assume that M is ﬁbered: M = Σ × [0, 1]/(x, 0) ∼ (φ(x), 1) L. Benard, J. Dubois, M. for some surface Σ and some diﬀeomorphism φ:Σ → Σ, called Heusener, J. Porti the monodromy. The representation ρn restricts to a representation of π1(Σ), and the monodromy acts on ρn,Σ by conjugation. We prove the following: Theorem (BDHP’19)

The action of the monodromy φ on [ρn,Σ] has hyperbolic dynamic. Namely its tangent map has no eigenvalues of modulus one.

A dynamical application of our ﬁrst theorem

Asymptotic of twisted polynomials Assume that M is fibered: M = Σ × [0, 1]/(x, 0) ∼ (φ(x), 1) L. Benard, J. Dubois, M. for some surface Σ and some diffeomorphism φ:Σ → Σ, called Heusener, J. Porti the monodromy. The representation ρn restricts to a representation of π1(Σ), and the monodromy acts on ρn,Σ by conjugation. In other words, [ρn,Σ] is a fixed point for the action of φ on the character variety of Σ. A dynamical application of our first theorem

Asymptotic of twisted polynomials Assume that M is fibered: M = Σ × [0, 1]/(x, 0) ∼ (φ(x), 1) L. Benard, J. Dubois, M. for some surface Σ and some diffeomorphism φ:Σ → Σ, called Heusener, J. Porti the monodromy. The representation ρn restricts to a representation of π1(Σ), and the monodromy acts on ρn,Σ by conjugation. In other words, [ρn,Σ] is a fixed point for the action of φ on the character variety of Σ. We prove the following: Theorem (BDHP’19)

The action of the monodromy φ on [ρn,Σ] has hyperbolic dynamic. Namely its tangent map has no eigenvalues of modulus one. What we prove is indeed a reﬁnement of the theorem of Weil: Proposition (BDHP’19)

The tangent map of φ acting on [ρn,Σ] has characteristic polynomial equal to the twisted Alexander polynomial.

We conclude with our ﬁrst theorem.

About the proof

Asymptotic of twisted polynomials

L. Benard, J. The proof comes from the following well-known fact theorem Dubois, M. Heusener, J. (due to Weyl): the tangent space of the character variety X (Σ) Porti naturally identiﬁes with the ﬁrst twisted cohomology group

1 T[ρn,Σ]X (Σ) ' H (Σ, Ad ◦ρn,Σ) We conclude with our ﬁrst theorem.

About the proof

Asymptotic of twisted polynomials

1 T[ρn,Σ]X (Σ) ' H (Σ, Ad ◦ρn,Σ)

What we prove is indeed a reﬁnement of the theorem of Weil: Proposition (BDHP’19)

The tangent map of φ acting on [ρn,Σ] has characteristic polynomial equal to the twisted Alexander polynomial. About the proof

Asymptotic of twisted polynomials

1 T[ρn,Σ]X (Σ) ' H (Σ, Ad ◦ρn,Σ)

What we prove is indeed a reﬁnement of the theorem of Weil: Proposition (BDHP’19)

The tangent map of φ acting on [ρn,Σ] has characteristic polynomial equal to the twisted Alexander polynomial.

We conclude with our ﬁrst theorem. Jensen’s formula relates the Mahler measure with the roots of P: X m(P) = log |ζ| P(ζ)=0 |ζ|>1 We obtain Theorem (BDHP’19)

m(∆n ) Vol(M) lim M = n→∞ n2 4π

An application of the main theorem.

Asymptotic of twisted We deﬁne the Mahler measure of a polynomial P as polynomials Z 1 L. Benard, J. 1 iθ Dubois, M. m(P) = log |P(e )|dθ. Heusener, J. 2π 0 Porti We obtain Theorem (BDHP’19)

m(∆n ) Vol(M) lim M = n→∞ n2 4π

An application of the main theorem.

Asymptotic of twisted We deﬁne the Mahler measure of a polynomial P as polynomials Z 1 L. Benard, J. 1 iθ Dubois, M. m(P) = log |P(e )|dθ. Heusener, J. 2π 0 Porti Jensen’s formula relates the Mahler measure with the roots of P: X m(P) = log |ζ| P(ζ)=0 |ζ|>1 An application of the main theorem.

m(∆n ) Vol(M) lim M = n→∞ n2 4π Asymptotic of twisted polynomials

L. Benard, J. Dubois, M. Heusener, J. Porti

Thank you!