TORUS COUNTING AND SELF-JOININGS OF KLEINIAN GROUPS SAM EDWARDS MINJU LEE AND HEE OH

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TORUS COUNTING AND SELF-JOININGS OF KLEINIAN

GROUPS

SAM EDWARDS, MINJU LEE, AND HEE OH

Abstract. For any d≥1, we obtain counting and equidistribution

results for tori with small volume for a class of d-dimensional torus

packings, invariant under a self-joining Γρ<Qd

i=1 PSL2(C) of a Kleinian

group Γ formed by a d-tuple of convex-cocompact representations ρ=

(ρ1,··· , ρd). More precisely, if Pis a Γρ-admissible d-dimensional torus

packing, then for any bounded subset E⊂Cdwith ∂E contained in a

proper real algebraic subvariety, we have

lim

s→0sδL1(ρ)·#{T∈ P : Vol(T)> s, T ∩E̸=∅} =cP·ωρ(E∩Λρ).

Here 0 < δL1(ρ)≤2/√dis the critical exponent of Γρwith respect to

the L1-metric on the product Qd

i=1 H3, Λρ⊂(C∪{∞})dis the limit set

of Γρ, and ωρis a locally ﬁnite Borel measure on Cd∩Λρwhich can be

explicitly described. The class of admissible torus packings we consider

arises naturally from the Teichm¨uller theory of Kleinian groups. Our

work extends previous results of Oh-Shah [24] on circle packings (i.e.

one-dimensional torus packings) to d-torus packings.

Contents

1. Introduction 2

2. Self-joinings and higher rank Patterson-Sullivan theory 8

3. Properties of admissible torus packings 11

4. Torus counting function for admissible torus packings 15

5. Mixing and equidistribution with uniform bounds 19

6. The measure ωψ21

7. Equidistribution in average. 23

8. Proof of the main counting theorem 26

9. PS-measures are null on algebraic varieties 33

References 35

Edwards’s work was supported by the Additional Funding Programme for Mathemati-

cal Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Math-

ematical Research. Oh is partially supported by the NSF grant No. DMS-1900101.

arXiv:2210.10229v2 [math.DS] 14 Nov 2023

2 SAM EDWARDS, MINJU LEE, AND HEE OH

1. Introduction

In this paper, we obtain counting and equidistribution results for a cer-

tain class of d-dimensional torus packings invariant under self-joinings of

Kleinian groups for any d≥1. One-dimensional torus packings are pre-

cisely circle packings. To motivate the formulation of our main results, we

begin by reviewing counting results for circle packings that are invariant

under Kleinian groups ([15], [23], [24], [25], [27], etc).

Circle counting. A circle packing in the complex plane Cis simply a

nonempty family of circles in C, for which we allow intersections among

themselves. In the whole paper, lines are also considered as circles of inﬁnite

radii. Let Γ <PSL2(C) = Isom+(H3) be a Zariski-dense convex-cocompact

discrete subgroup. We call a circle packing PΓ-admissible if

• P consists of ﬁnitely many Γ-orbits of circles;

• P is locally ﬁnite, in the sense that no inﬁnite sequence of circles in

Pconverges to a circle.

We denote by 0 < δΓ≤2 the critical exponent of Γ i.e. the abscissa of

convergence for the Poincare series P(s) := Pg∈Γe−sdH3(gp,p)where p∈H3

is any point and dH3is the hyperbolic metric so that (H3,dH3) has con-

stant curvature −1. The extended complex plane ˆ

C=C∪ {∞} can be

regarded as the geometric boundary of H3. The limit set of Γ is the set of

all accumulation points of the orbit Γ(z) of z∈ˆ

C; we denote it by ΛΓ⊂ˆ

Theorem 1.1. [24] For any Γ-admissible circle packing P, there exists a

constant cP>0such that for any bounded measurable subset E⊂Cwhose

boundary is contained in a proper real algebraic subvariety of C,

lim

s→0sδΓ#{C∈ P : radius(C)≥s, C ∩E̸=∅} =cPωΓ(E∩ΛΓ);

here ωΓis the δΓ-dimensional Hausdorﬀ measure on C∩ΛΓwith respect to

the Euclidean metric on C.

This theorem holds for a more general class of circle packings invariant by

geometrically ﬁnite Kleinian groups, which includes the famous Apollonian

circle packings for which the relevant counting result was ﬁrst obtained in

[15] (see [24] for more details and examples).

Torus counting. The main goal of this paper is to prove a higher dimen-

sional analogue of Theorem 1.1. Let d≥1. By a torus in Cdwe mean a

Cartesian product of d-number of circles C1,··· , Cd⊂C. However, it will

be convenient to consider it as a d-tuple of circles

T= (C1,··· , Cd) (1.2)

rather than a subset C1× ··· × Cd⊂Cd. A d-dimensional torus packing

in Cdis simply a nonempty family of d-tori in Cd.

TORUS COUNTING 3

The volume of Tis given by

Vol(T) =

i=1

2πradius Ci.

Figure 1 shows some image of a 2-torus packing. Although the torus

T=C1×C2in Fig. 1 appears to be in R3, it should be understood as a

subset of R4, representing the Cartesian product of the boundary circles of

two discs.

Figure 1. A torus packing

We are interested in understanding the asymptotic counting and distri-

bution of tori with small volumes in a torus packing that is invariant under

a self-joining of a convex-cocompact Kleinian group.

Let Γ <PSL2(C) be a convex-cocompact discrete subgroup and ρ= (ρ1=

id, ρ2,··· , ρd) be a d-tuple of faithful convex-cocompact representations of

Γ into PSL2(C). Let G=Qd

i=1 PSL2(C). The self-joining of Γ via ρis

deﬁned as the following discrete subgroup of G:

Γρ={ρ1(g),··· , ρd(g):g∈Γ}.

Throughout the paper we will always assume that Γρis Zariski-dense in

G. Each ρiinduces a unique equivariant homeomorphism fi: ΛΓ→Λρi(Γ),

which is called the ρi-boundary map [36]. In this paper, we deﬁne the limit

set of Γρby

Λρ={(f1(ξ),··· , fd(ξ)) ∈ˆ

Cd:ξ∈ΛΓ}.

We call a torus T= (C1,··· , Cd) Γρ-admissible if for each 1 ≤i≤d,

•ρi(Γ)Ciis a locally ﬁnite circle packing;

•fi(C1∩ΛΓ) = Ci∩Λρi(Γ).

The second condition is equivalent to

T∩Λρ={(ξ1,··· , ξd)∈Λρ:ξ1∈C1∩ΛΓ},

that is, the circular slice C1∩ΛΓcompletely determines the toric slice T∩Λρ.

4 SAM EDWARDS, MINJU LEE, AND HEE OH

Deﬁnition 1.3. A torus packing Pis called Γρ-admissible if

• P consists of ﬁnitely many Γρ-orbits of Γρ-admissible tori;

• P is locally ﬁnite in the sense that no inﬁnite sequence of tori in P

converges to a torus.

Remark 1.4. We remark that when #(C1∩ΛΓ)≥3, the locally ﬁniteness

hypotheses in the above deﬁnition can be reduced to the local-ﬁniteness of

the circle packing ΓC1(see Prop. 3.11).

We denote by δL1(ρ) the abscissa of convergence of the series

s7→ PL1(s) := X

g∈Γ

e−sPd

i=1 dH3(ρi(g)p,p)

for p∈H3, which is the critical exponent of Γρwith respect to the L1

product metric on Qd

i=1(H3,dH3).

We ﬁrst state the following special case of the main result of this paper.

Theorem 1.5. Let Pbe a Γρ-admissible torus packing. There exists a

constant cP>0such that for any bounded measurable subset E⊂Cdwith

boundary contained in a proper real algebraic subvariety, we have

lim

s→0sδL1(ρ)#{T∈ P : Vol(T)> s, T ∩E̸=∅} =cPωΓρ(E∩Λρ),

where ωΓρis a locally ﬁnite Borel measure on Cd∩Λρwhich can be explicitly

described. In particular, if Pis bounded, then

lim

s→0sδL1(ρ)#{T∈ P : Vol(T)> s}=cP|ωΓρ|.

Remark 1.6. (1) Since δL1(ρ) is bounded above by the usual critical

exponent δΓρof Γρwith respect to the Riemannian metric (which

equals the L2product metric) on Qd

i=1 H3, we have

0< δL1(ρ)≤δΓρ≤1

√dmax

i(dim(Λρi(Γ))) ≤2

√d

by [13, Coro. 3.6]; here the notation dim(·) means the Hausdorﬀ

dimension of a measurable subset of ˆ

C≃S2with respect to the

spherical metric.

(2) If all ρi: Γ →PSL2(C) are quasiconformal deformations of Γ and

∞/∈ ∪d

i=1Λρi(Γ), then for any bounded torus packing P= ΓρT

with T= (C1,··· , Cd), Pis locally ﬁnite if and only if {ρi(γ)Ci:

γ∈Γ}is a locally ﬁnite circle packing for all 1 ≤i≤d. This

is because the boundary map fiis the restriction to Λρi(Γ) of the

quasiconformal homeomorphism Fi:ˆ

C→ˆ

Cassociated to ρi, and

under the hypothesis ∞/∈ ∪d

i=1Λρi(Γ), the Fiare bi-H¨older maps on

any compact subset of C([7], [36]).

TORUS COUNTING 5

More general torus-counting theorems. In order to present a more

general torus-counting theorem, we deﬁne the length vector of a torus T=

(C1,··· , Cd) by

v(T) = −log radius(C1),··· ,log radius(Cd)∈Rd;

where we used the negative sign so that the i-th coordinate of v(T) tends

to +∞as Cishrinks to a point. The following result is the main theorem

of this paper.

Theorem 1.7. Let ψbe any linear form on Rdsuch that ψ > 0on (R≥0)d−

{0}. There exist δψ>0and a locally ﬁnite Borel measure ωψon Λρ∩

Cddepending only on Γρand ψfor which the following hold: for any Γρ-

admissible torus packing P, there exists a constant cψ=cP,ψ >0such that

for any bounded measurable subset E⊂Cdwith boundary contained in a

proper real algebraic subvariety, we have, as R→ ∞,

lim

R→∞

eδψR#{T∈ P :ψ(v(T)) < R, T ∩E̸=∅} =cψωψ(E∩Λρ).(1.8)

The description of the measure ωψ(Def. 6.1) depends on the higher rank

Patterson-Sullivan theory. In fact, it is equivalent to the unique (Γρ, ψ0)-

conformal measure on Λρ, where ψ0is the unique Γρ-critical linear form

(Def. 2.8) proportional to ψ. We refer to Def. 2.6 for the deﬁnition of δψ.

Remark 1.9. (1) Theorem 1.5 can be deduced from this theorem by

considering the linear form ψ: (t1,··· , td)7→ t1+··· +td(see Ex.

8.3).

(2) Our approach can also handle the case where ψ(v(T)) is replaced by

the Euclidean norm of v(T) in (1.8); indeed, the analysis involved in

that case is easier due to the strict convexity of the Euclidean balls

in Rd(see the last subsection of Sec 8).

(3) The fact that the sublevel sets {t∈Rd:ψ(t)< c}are linear (hence

not strictly convex) presents new technical diﬃculties which were

not dealt with in related previous works such as [24] and [5].

We now discuss examples of admissible torus packings arising naturally

from the Teichm¨uller theory of Kleinian groups.

Example 1.10. (1) Let Γ <PSL2(C) be a Zariski-dense and convex-

cocompact subgroup whose domain of discontinuity ΩΓ:= ˆ

C−ΛΓ

has a connected component which is a round open disk B. Let

C1:= ∂B and d≥2. By the Teichm¨uller theory of Γ, which relates

the Teichm¨uller space of the Riemann surface Γ\ΩΓand the quasi-

conformal deformation space of Γ ([20, Thm. 5.27], [19]) we may

choose quasi-conformal deformations ρi: Γ →PSL2(C), 2 ≤i≤d,

whose associated quasiconformal maps fi:ˆ

C→ˆ

Cmap C1to a

circle, say, Ci. Then T= (C1,··· , Cd)isaΓρ-admissible torus for

ρ= (id, ρ2,··· , ρd) and hence P= ΓρTis a Γρ-admissible torus

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TORUSCOUNTINGANDSELF-JOININGSOFKLEINIANGROUPSSAMEDWARDS,MINJULEE,ANDHEEOHAbstract.Foranyd≥1,weobtaincountingandequidistributionresultsfortoriwithsmallvolumeforaclassofd-dimensionaltoruspackings,invariantunderaself-joiningΓρ

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TORUS COUNTING AND SELF-JOININGS OF KLEINIAN GROUPS SAM EDWARDS MINJU LEE AND HEE OH

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