Representations of discrete groups

Chapter III: Infinite energy harmonic maps and AdS 3-manifolds

3.2 Representations of discrete groups

Smillie for their interest and helpful conversations, as well as François Guéritaud for graciously answering some questions over email. Finally, I would like to thank my friend Arian Jadbabaie for helping create the figure on Inkscape.

3.2 Representations of discrete groups

Optimal Lipschitz constants

GivenΓdiscrete, 𝜌 :Γ →Isom(𝑋 , 𝑔), and 𝑗 : Γ→ PSL₂(R) geometrically finite, we set

𝐶(𝑗 , 𝜌) :=inf Lip(𝑓),

where the infimum is taken over the family of all(𝑗 , 𝜌)-equivariant Lipschitz maps.

The theorem below is Theorem 1.8 in [GK17].

Theorem 3.2.2. (Guéritaud, Kassel) Let Γ be a discrete group and 𝜌, 𝑗 : Γ → PSL₂(R) two representations with 𝑗 geometrically finite. Then𝐶(𝑗 , 𝜌) < 1if and only if

𝐶(𝑗 , 𝜌)^′:=sup

ℓ(𝜌(𝛾)) ℓ(𝑗(𝛾)) <1,

unless𝜌has exactly one fixed point on𝜕_∞Hand there exists a𝛾 ∈ Γsuch that 𝑗(𝛾) is parabolic and 𝜌(𝛾) is not elliptic.

Remark 3.2.3. As we will see later, equivariant harmonic maps only exist for reductive representations. To dominate non-reductive representations we would like to use a version of Theorem 3.2.2 that holds for variable curvature. The result and the proof of Theorem 3.2.2 do not directly transfer, and trying to extend them is outside the scope of the current work. Hence, for the non-reductive case we settle for length spectrum domination, although we expect the full domination result to be true. From the theorem above, non-reductive representations still lead to AdS 3-manifolds, which is the most important application.

Now suppose Σ = Σ/Γ˜ is a complete finite volume hyperbolic orbifold. By the Selberg lemma, Γ admits a finite index torsion free normal subgroup Γ₀. The quotient ˜Σ/Γ₀ is a complete finite volume hyperbolic manifold. We close this section with a lemma that reduces Theorem 3B to the case of hyperbolic manifolds.

Lemma 3.2.4. LetΓbe a discrete group andΓ₀⊂ Γa finite index normal subgroup.

Let 𝜌 : Γ → Isom(𝑋 , 𝑔) and 𝑗 : Γ → PSL₂(R) be representations and let 𝜌

0and 𝑗0be their restrictions toΓ₀. Then𝐶(𝑗 , 𝜌) =𝐶(𝑗

0, 𝜌

0).

This is essentially done in [GK17], although the authors prove something more general and restrict to the case 𝑋 = H^𝑛. For the convenience of the reader, we essentially repeat the proof. We use a lemma from [GK17].

Lemma 3.2.5. Let 𝐼 be any countable index set and 𝛼 = (𝛼_𝑖)_𝑖∈𝐼 ⊂ R a sequence summing to1. Given 𝑝 ∈𝐾 ⊂ Hand 𝑓_𝑖 :𝐾 → 𝑋,𝑖 ∈ 𝐼 such that

∑︁

𝑖∈𝐼

𝛼_𝑖𝑑(𝑓

1(𝑝), 𝑓_𝑖(𝑝)) < ∞, the map

𝑓 :=∑︁

𝑖∈𝐼

𝛼_𝑖𝑓_𝑖, 𝑥 ↦→argminn

𝑝^′∈ 𝑋 :

∑︁

𝑖∈𝐼

𝛼_𝑖𝑑(𝑝^′, 𝑓_𝑖(𝑥)) < ∞o is well-defined and satisfies

Lip𝑥(𝑓) ≤∑︁

𝑖

𝛼_𝑖Lip𝑥(𝑓_𝑖), Lip𝑌(𝑓) ≤∑︁

𝑖

𝛼_𝑖Lip𝑌(𝑓_𝑖).

If each 𝑓_𝑖 is equivariant with respect to a pair of representations then so is 𝑓. The authors give a proof for 𝑋 = H^𝑛 but the proof only uses the fact that H^𝑛 is a CAT(0)metric space.

Proof of lemma 3.2.4. If no(𝑗^′, 𝜌^′)-equivariant maps exist there is nothing to prove, so assume otherwise. The inequality 𝐶(𝑗^′, 𝜌^′) ≤ 𝐶(𝑗 , 𝜌) is obvious because any (𝑗 , 𝜌)-equivariant map is(𝑗^′, 𝜌^′)-equivariant. As for the other inequality, write

Γ =

𝑟

Þ

𝑖=1

𝛾_𝑖Γ₀

for some collection of coset representatives𝛾_𝑖. Let 𝑓 be a (𝑗^′, 𝜌^′)-equivariant map.

Notice that for any𝛾 ∈Γ, the map

𝑓_𝛾 := 𝜌(𝛾)⁻¹◦ 𝑓 ◦ 𝑗(𝛾)

depends only on the coset𝛾Γ₀. Indeed, suppose we are given𝛾

1, 𝛾

2 ∈ Γsuch that 𝛾1𝛾⁻¹

2 ∈ Γ₀. For𝑥 ∈Hlet𝑦= 𝑗(𝛾

2)⁻¹𝑥. Then 𝑓_𝛾

1(𝑥)= 𝜌(𝛾

1)⁻¹◦ 𝑓(𝑗(𝛾

1𝛾⁻¹

2 )𝑦)= 𝜌(𝛾

2)⁻¹◦ 𝑓(𝑦) = 𝑓_𝛾

2(𝑥). By Lemma 3.2.5 the map

𝑓^′:=

𝑟

∑︁

𝑖=1

1 𝑟

· 𝑓_𝛾

𝑖

satisfies

𝜌(𝛾)⁻¹◦ 𝑓^′◦ 𝑗(𝛾) =

𝑟

∑︁

𝑖=1

1 𝑟

· 𝑓_{𝛾 𝛾}

𝑖 = 𝑓^′

since the sum in the middle is just a rearrangement of the sum describing 𝑓^′. By Lemma 3.2.5 again we have Lip(𝑓^′) ≤ Lip(𝑓). Taking Lip(𝑓) → 𝐶(𝑗^′, 𝜌^′), the

lemma follows. □

Useful lemmas

We collect some general results on harmonic maps that we’ll use throughout.

Theorem 3.2.6(Ishihara). Suppose that all sectional curvatures of a manifold(𝑋 , 𝜈) are non-negative. Then 𝑓 : (Σ, 𝜇) → (𝑋 , 𝜈) is harmonic if and only if it pulls back germs of convex functions to germs of subharmonic functions.

We record a corollary.

Corollary 3.2.7. Suppose that all sectional curvatures of a manifold (𝑋 , 𝜈) are non-negative and let 𝑓

1, 𝑓

2 : (Σ, 𝜇) → (𝑋 , 𝜈) be harmonic maps. Let 𝑑_𝜈 be the Riemannian distance function on (𝑋 , 𝜈). Then the function on Σ given by 𝑝 ↦→

𝑑(𝑓

1(𝑝), 𝑓

2(𝑝))is subharmonic.

A harmonic function between Euclidean spaces has a representation in terms of the Poisson integral formula. Out of this formula, one can obtain local𝐶^𝑘 bounds in terms of local𝐶⁰bounds. For harmonic maps between manifolds, Cheng’s lemma gives𝐶1bounds in terms of𝐶0control.

Lemma 3.2.8(Cheng’s lemma). Let𝑋 and𝑌 be Hadamard manifolds with−𝑏2 ≤ 𝐾_𝑋 ≤0anddim𝑋 = 𝑘 .Let𝑧 ∈ 𝑋,𝑟 >0,and letℎ :𝐵(𝑥 , 𝑟) →𝑌be a𝐶^∞harmonic map such that the imageℎ(𝐵(𝑧, 𝑟))is contained in a ball of radius𝑅

0.Then

||𝐷 ℎ(𝑧) || ≤2⁵𝑘1+𝑏𝑟 𝑟

𝑅0.

See [Che80]. Local 𝐶^𝑘 bounds are deduced from local 𝐶¹ bounds via elliptic bootstrapping.

Energy of harmonic maps

We prove some preliminary results relevant to equivariant harmonic maps from surfaces with punctures.

Proposition 3.2.9. IfΣ =Σ/Γ˜ is a complete finite volume hyperbolic surface,(𝑋 , 𝑔) is a Hadamard manifold, and 𝜌 :Γ→ Isom(𝑋 , 𝑔) is a representation, then a finite energy 𝜌-equivariant map exists if and only if𝜌has no hyperbolic monodromy.

Before we begin, we modify the metric to a new one that will be used throughout the chapter. Label the cusp neighbourhoods 𝐶

1, . . . , 𝐶_𝑛. Take collar neighbourhoods 𝑈_𝑘 of each 𝜕𝐶_𝑘 inside Σ\𝐶_𝑘 and consider the metric on Σ that agrees with the

hyperbolic metric on Σ\(∪𝑘𝐶_𝑘) and is flat on each𝐶_𝑘 ∪𝑈_𝑘. Then interpolate on a neighbourhood of 𝜕𝑈_𝑘\𝜕𝐶_𝑘 that does not touch 𝜕𝐶_𝑘 to a smooth non-positively curved metric𝜎^′, conformally equivalent to the hyperbolic metric. We will call this the flat-cylinder metric.

We also take this opportunity to introduce thetransverse horospherical flow. With 𝑋 as above, consider a horoball 𝐵 ⊂ 𝑋 with horospherical boundary 𝐻 centered at the fixed point𝜉of a parabolic isometry𝜓. The subgroup generated by𝜓 preserves 𝐻 and𝐵. The data (𝐵, 𝐻 , 𝜉)determines a flow𝜑_𝑡 :𝐵× [0,∞) → 𝐵defined by

𝜑_𝑡(𝑝) =𝛼_𝑝,𝜉(𝑡),

where𝛼_𝑝,𝜉 : [0,∞) → 𝑋is the unique geodesic starting from𝑝and tending towards 𝜉 at∞.

Lemma 3.2.10. The transverse horospherical flow is⟨𝜓⟩-equivariant.

Proof. Notice

𝛼_{𝜓·𝑝,𝜉}(0) =𝜓· 𝑝 =𝜓·𝛼_𝑝,𝜉(0).

Since𝛼_{𝜓·𝑝,𝜉}(𝑡) and𝜓·𝛼_𝑝,𝜉(𝑡) describe geodesics with the same starting point and

end point, they are identical. □

Proof of proposition 3.2.9. By conformal invariance of energy we’re permitted to do all of our computations in the flat-cylinder metric. Firstly let us assume there is a peripheral𝛾 such that 𝜌(𝛾) is hyperbolic. Take any equivariant map 𝑓 : ˜Σ → 𝑋 and fix a cusp neighbourhood associated to the peripheral and isometric to𝑈(𝜏). As𝜌(𝛾)is hyperbolic,

𝑑_𝑔(𝑓(𝑖 𝑦), 𝑓(𝜏+𝑖 𝑦)) =𝑑_𝑔(𝑓(𝑖 𝑦), 𝜌(𝛾)𝑓(𝑖 𝑦)) ≥ ℓ(𝜌(𝛾)) > 0,

independent of 𝑦. For each 𝑦 let 𝛾_𝑦 be the path 𝑥 ↦→ 𝑓(𝑥+𝑖 𝑦), 𝑥 ∈ [0, 𝜏]. The inequality above implies

ℓ(𝜌(𝛾)) ≤

∫ 𝜏 0

||𝑑 𝛾_𝑦||𝜎^′𝑑𝑦 and by Cauchy-Schwarz we obtain

ℓ(𝜌(𝛾))² 2𝜏

≤ 1 2

∫ ^𝜏

0

||𝑑 𝛾_𝑦||²_𝜎′𝑑𝑦 ≤

∫ ^𝜏

0

𝑒(𝑓) (𝑥 , 𝑦)𝑑𝑦 .

Hence,

𝐸(𝑓) ≥𝐸_𝑉(𝑓) =

∫ ∞

𝑎

∫ ^𝜏

𝑎

𝑒(𝑓) (𝑥 , 𝑦)𝑑𝑥 𝑑𝑦 ≥ ℓ(𝜌(𝛾))² 2𝜏

∫ ∞

𝑎

𝑑𝑦 =∞, which shows all equivariant maps have infinite energy.

For the other direction, we simply produce an equivariant finite energy map. We build a finite energy map in a neighbourhood of each cusp, equivariant with respect to the subgroup generated by 𝜌(𝛾_𝑗) and then extend smoothly to a 𝜌-equivariant map on the (compact) complement of the cusps.

By induction it suffices to assume that there is only one cusp neighbourhood𝑉. We identify it with some𝑈(𝜏). Let 𝛾 be the corresponding curve. If 𝜌(𝛾) is elliptic then we simply map all of𝑉 to a fixed point of𝜌(𝛾). This is clearly equivariant and has zero energy in𝑉. Henceforward we assume𝜌(𝛾)is parabolic. ⟨𝜌(𝛾)⟩stabilizes a horoball 𝐵 with horopsherical boundary 𝐻. Let 𝑔 be any 𝐶^∞ 𝜌|_⟨𝛾⟩-equivariant mapR→ 𝐻. Define 𝑓 : ˜𝑉 → 𝐵by

𝑓(𝑥+𝑖 𝑦) =𝜑_𝑣

log(𝑦+1)(𝑔(𝑥)),

where𝜑is the transverse horospherical flow with respect to the fixed point and𝑣 >0 will be specified later. We compute

|𝑑 𝑓(𝜕/𝜕 𝑦) |𝑓(𝑥+𝑖 𝑦) =|𝜕/𝜕 𝑦(𝑣log(𝑦+1)) | = 𝑣 𝑦+1

. Next, note that

𝐽_𝑥(𝑦) := 𝜕

𝜕 𝑥

𝑓(𝑥+𝑖 𝑦)

is a Jacobi field for each𝑥. By the curvature assumption on𝑋, the Rauch comparison theorem shows that any Jacobi field on 𝑋 along a geodesic decays exponentially in time: there is a𝑢 >0 such that

|𝐽_𝑥(𝑦) | ≤ 𝐴𝑒^{−𝑢·𝑣log(𝑦+1)} for all𝑥. Now choose𝑣 so that𝑢𝑣 ≥ 1. Then

|𝑑 𝑓(𝜕/𝜕 𝑥) |𝑓(𝑥+𝑖 𝑦) ≤ 𝐴 (𝑦+1)^𝑢𝑣, and furthermore

𝐸_𝑉(𝑓) ≤ 1 2

∫ ∞

0

∫ 𝜏 0

𝑣²+ 𝐴²

(𝑦+1)²𝑑𝑥 𝑑𝑦 = 𝜏 2

(𝑣²+ 𝐴²) < ∞,

and the result follows. □

Remark 3.2.11. The total energy of a harmonic map is finite if and only if the Hopf differential is integrable. Passing to polar coordinates, we see that an integrable holomorphic quadratic differential has a pole of order at most 1 at a puncture.

Suppose a representation admits a finite energy equivariant map. If it does not fix a point on the ideal boundary, the harmonic map determined by Theorem 2.4.4 is unique. If 𝜌 stabilizes a geodesic, there is a 1-parameter family of harmonic maps that differ by translations along that geodesic axis. The standard methods push through to give a uniqueness criterion in our setting.

Lemma 3.2.12. LetΣbe a complete finite volume hyperbolic surface, let(𝑋 , 𝑔)be Hadamard, and let 𝑓

1 and 𝑓

2equivariant harmonic maps for 𝜌 such that the map 𝑧 ↦→ 𝑑(𝑓

1, 𝑓

2) (𝑧) is bounded. If 𝜌 does not fix a point on 𝜕_∞𝑋 then 𝑓

1 = 𝑓

2. If 𝜌 stabilizes a geodesic, then 𝑓

1and 𝑓

2may differ by translation along a geodesic.

Proof. For𝑧 ∈ Σ let {𝑒

1, 𝑒

2} be an orthonormal frame for the tangent bundle in a neighbourhood of 𝑧 and let {𝑣⁰

1, . . . , 𝑣⁰

𝑛}, {𝑣¹

1, . . . , 𝑣¹

𝑛} be orthonormal frames for neighbourhoods of 𝑓

1(𝑧), 𝑓

2(𝑧)respectively. In these frames we write (𝑓_𝑘)_∗𝑒_𝑖 =

𝑛

∑︁

𝑚=1

𝜆^𝑘

𝑖,𝑚𝑣^𝑘

𝑚. {𝑣0

1

, . . . , 𝑣0 𝑛, 𝑣1

1, . . . , 𝑣1

𝑛}is an orthonormal frame near(𝑓

1(𝑧), 𝑓

2(𝑧)) ∈ 𝑋×𝑋. Define vector fields 𝑋_𝑖 ∈ Γ(𝑇(𝑋 × 𝑋)) so that around (𝑓

1(𝑧), 𝑓

2(𝑧)) the projections onto the first and second factors are 𝑓^∗

1𝑒_𝑖 and 𝑓^∗

2𝑒_𝑖 respectively. Let 𝑑 : ˜Σ → R be the function

𝑑(𝑧) =𝑑_𝑔⊕𝑔(𝑓

1(𝑧), 𝑓

2(𝑧))

which is𝐶^∞away from the diagonal. From a computation in [SY97, Chapter 11.2], if we assume 𝑓

1(𝑧) ≠ 𝑓

2(𝑧)then from the fact that the 𝑓_𝑘 are harmonic, Δ𝑑² ≥ 2𝑑

∑︁2

𝑖=1

𝐷²𝑑_𝑔(𝑋_𝑖, 𝑋_𝑖)

around𝑧. Above,Δis the Laplacian on ˜Σand𝐷²𝑑_𝑔is the Hessian of𝑑_𝑔, the distance function on (𝑋 , 𝑔).

By equivariance, 𝑑 descends to a bounded subharmonic function on Σ. As Σ is parabolic in the potential theoretic sense, this function is constant. Therefore,

2𝑑

∑︁2

𝑖=1

𝐷²𝑑_𝑔(𝑋_𝑖, 𝑋_𝑖) =0.

This forces 𝑑 = 0 or 𝐷²𝑑_𝑔(𝑋_𝑖, 𝑋_𝑖) = 0. In the first case we have 𝑓

1 = 𝑓

2 so let us move to the latter. From an argument in [SY97, Chapter 11.2], this implies either 𝑓

1 = 𝑓

2 or 𝑓

1 and 𝑓

2 have image in a geodesic and differ by a translation along that geodesic. By equivariance, this last case can only occur if 𝜌 stabilizes a

geodesic. □

3.3 Energy domination