Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface Sݑ†Sitalic_S and a certain unipotent flow on the product space T1(S)×T1(S)subscriptݑ‡1ݑ†subscriptݑ‡1ݑ†T_1(S)\times T_1(S)italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) × italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ), we obtain a local asymptotic equidistribution result for long closed geodesics on Sݑ†Sitalic_S. Applications include asymptotic estimates for the number of pants immersions into Sݑ†Sitalic_S satisfying various geometric constraints. Also we show that two closed geodesics γ1,γ2subscriptݛ¾1subscriptݛ¾2\gamma_1,\gamma_2italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of length close to Lݐ¿Litalic_L chosen uniformly at random have a high probability of partially bounding an immersed 4444-holed sphere whose other boundary components also have length close to Lݐ¿Litalic_L.

MSC: 20E09, 20F69, 37E35, 51M10 Keywords: hyperbolic surface, pants, surface subgroup, subgroup growth, equidistribution.

One motivation for this work came from two well-known conjectures:

Conjecture 1.1.

(The Surface Subgroup Conjecture) Let ℳℳ\cal Mcaligraphic_M be a closed hyperbolic 3333-manifold. Then there exists a π1subscriptݜ‹1\pi_1italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-injective map j:S→ℳnormal-:ݑ—normal-→ݑ†ℳj:S\to\cal Mitalic_j : italic_S → caligraphic_M from a closed surface Sݑ†Sitalic_S of genus at least 2 into ℳℳ\cal Mcaligraphic_M.

Conjecture 1.2.

(The hyperbolic Ehrenpreis conjecture)([Ehrenpreis],[Gendron]) Let ϵ>0italic-ϵ0\epsilon>0italic_ϵ >0 and let S1,S2subscriptݑ†1subscriptݑ†2S_1,S_2italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be two closed hyperbolic surfaces. Then there exists finite-sheeted locally isometric covers S~isubscriptnormal-~ݑ†ݑ–\tildeS_iover~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of Sisubscriptݑ†ݑ–S_iitalic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (for i=1,2ݑ–12i=1,2italic_i = 1 , 2) such that there is a (1+ϵ)1italic-ϵ(1+\epsilon)( 1 + italic_ϵ ) bi-Lipschitz homeomorphism between S~1subscriptnormal-~ݑ†1\tildeS_1over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and S~2subscriptnormal-~ݑ†2\tildeS_2over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

Attempting to understand these conjectures led to the study of immersions of three-holed spheres into 3333-manifolds ℳℳ\cal Mcaligraphic_M and cross products S1×S2subscriptݑ†1subscriptݑ†2S_1\times S_2italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT; we would like to glue such immersions together to obtain an immersion of a closed surface S→ℳ→ݑ†ℳS\to\cal Mitalic_S → caligraphic_M or S→S1×S2→ݑ†subscriptݑ†1subscriptݑ†2S\to S_1\times S_2italic_S → italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with tight control on its geometric structure.

But the simpler case, that of immersions of three-holed spheres into a closed hyperbolic surface Sݑ†Sitalic_S, is not well-understood. For instance, the following question is unknown.

Question 1.3.

Given ϵ>0italic-ϵ0\epsilon>0italic_ϵ >0, for sufficiently large Lݐ¿Litalic_L, does there exist a finite sheeted cover π:S~→Snormal-:ݜ‹normal-→normal-~ݑ†ݑ†\pi:\tildeS\to Sitalic_π : over~ start_ARG italic_S end_ARG → italic_S such that S~normal-~ݑ†\tildeSover~ start_ARG italic_S end_ARG admits a pair of pants decomposition, every geodesic of which has length in (L-ϵ,L+ϵ)ݐ¿italic-ϵݐ¿italic-ϵ(L-\epsilon,L+\epsilon)( italic_L - italic_ϵ , italic_L + italic_ϵ )?

Another motivation for the present work comes from a desire to “bridge the gap” between two research areas: group growth and subgroup growth. The former concerns itself with the asymptotic number of elements in a given group of word length less than Rݑ…Ritalic_R, the latter with the asymptotic number of subgroups with finite index less than Rݑ…Ritalic_R. Is there something in between? Among other things, here we study the asymptotic number of conjugacy classes of 2-generator subgroups of a surface group satisfying certain geometric conditions with the aim (not yet realized) of amalgamating these subgroups together to obtain finite index subgroups with geometric constraints.

1.1 Equidistibution Results

Let Sݑ†Sitalic_S be a fixed closed hyperbolic surface. Regard Sݑ†Sitalic_S as the quotient space ℍ2/Γsuperscriptℍ2Γ\mathbbH^2/\Gammablackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / roman_Γ where ΓΓ\Gammaroman_Γ is a fixed lattice in the group Isom+(ℍ2)ݐ¼ݑ ݑœsuperscriptݑšsuperscriptℍ2Isom^{+}(\mathbbH^2)italic_I italic_s italic_o italic_m start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) (=PSL2(ℝ))absentݑƒݑ†subscriptݐ¿2ℝ(=PSL_2(\mathbbR))( = italic_P italic_S italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_R ) ) of all orientation-preserving isometries of the hyperbolic plane ℍ2superscriptℍ2\mathbbH^2blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Let wݑ¤witalic_w be an arbitrary unit vector in the unit tangle bundle T1(S)subscriptݑ‡1ݑ†T_1(S)italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ). Consider the geodesic segment of length Lݐ¿Litalic_L tangent to wݑ¤witalic_w with wݑ¤witalic_w based at its midpoint. If the tangent vectors e1,e2subscriptݑ’1subscriptݑ’2e_1,e_2italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT at the endpoints are close then a short segment can be adjoined to it to obtain a closed path in Sݑ†Sitalic_S. The closed geodesic γݛ¾\gammaitalic_γ in the homotopy class of this path is very close to the original segment. A calculation we will use often quantifies how close. For example, we show that there is a function F=(F1,F2,F3)ݐ¹subscriptݐ¹1subscriptݐ¹2subscriptݐ¹3F=(F_1,F_2,F_3)italic_F = ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) of the position of e1subscriptݑ’1e_1italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT relative to e2subscriptݑ’2e_2italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT such that the distance from wݑ¤witalic_w to γݛ¾\gammaitalic_γ along a geodesic segment orthogonal to wݑ¤witalic_w equals F1e-L/2+O(e-L)subscriptݐ¹1superscriptݑ’ݐ¿2ݑ‚superscriptݑ’ݐ¿F_1e^-L/2+O(e^-L)italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_L / 2 end_POSTSUPERSCRIPT + italic_O ( italic_e start_POSTSUPERSCRIPT - italic_L end_POSTSUPERSCRIPT ), the angle at which this segment intersects γݛ¾\gammaitalic_γ equals π/2+F2e-L/2+O(e-L)ݜ‹2subscriptݐ¹2superscriptݑ’ݐ¿2ݑ‚superscriptݑ’ݐ¿\pi/2+F_2e^-L/2+O(e^-L)italic_π / 2 + italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_L / 2 end_POSTSUPERSCRIPT + italic_O ( italic_e start_POSTSUPERSCRIPT - italic_L end_POSTSUPERSCRIPT ) and the length of γݛ¾\gammaitalic_γ equals L+F3+O(e-L)ݐ¿subscriptݐ¹3ݑ‚superscriptݑ’ݐ¿L+F_3+O(e^-L)italic_L + italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_O ( italic_e start_POSTSUPERSCRIPT - italic_L end_POSTSUPERSCRIPT ). Roughly speaking, if the distance between e1subscriptݑ’1e_1italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and e2subscriptݑ’2e_2italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is less than ϵitalic-ϵ\epsilonitalic_ϵ then |F1|,|F2|,|F3|subscriptݐ¹1subscriptݐ¹2subscriptݐ¹3|F_1|,|F_2|,|F_3|| italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | , | italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | , | italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | are all less than ϵitalic-ϵ\epsilonitalic_ϵ as well. For a precise statement see corollaries 3.4 and 4.2 below.

If the vectors e1subscriptݑ’1e_1italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and e2subscriptݑ’2e_2italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are not close then we push wݑ¤witalic_w to its right along an orthogonal geodesic. The pair (e1,e2)subscriptݑ’1subscriptݑ’2(e_1,e_2)( italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) moves by a product of hypercycle flows in the product space T1(S)×T1(S)subscriptݑ‡1ݑ†subscriptݑ‡1ݑ†T_1(S)\times T_1(S)italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) × italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ). Pushing wݑ¤witalic_w a distance O(e-L/2)ݑ‚superscriptݑ’ݐ¿2O(e^-L/2)italic_O ( italic_e start_POSTSUPERSCRIPT - italic_L / 2 end_POSTSUPERSCRIPT ) amounts to flowing this pair for O(1)ݑ‚1O(1)italic_O ( 1 ) time. As Lݐ¿Litalic_L tends to infinity, this product hypercycle flow converges to a product of horocycle flows. As a consequence of Ratner’s work on Raghunathan’s conjectures we show that if the commensurator, Comm(Γ)ݐ¶ݑœݑšݑšΓComm(\Gamma)italic_C italic_o italic_m italic_m ( roman_Γ ), contains only orientation preserving isometries then the latter flow is uniformly equidistributed on the product space. By definition, Comm(Γ)ݐ¶ݑœݑšݑšΓComm(\Gamma)italic_C italic_o italic_m italic_m ( roman_Γ ) is the set of all isometries g∈Isom(ℍ2)ݑ”ݐ¼ݑ ݑœݑšsuperscriptℍ2g\in Isom(\mathbbH^2)italic_g ∈ italic_I italic_s italic_o italic_m ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) such that gΓg-1∩Γݑ”Γsuperscriptݑ”1Γg\Gamma g^-1\cap\Gammaitalic_g roman_Γ italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∩ roman_Γ has finite index in ΓΓ\Gammaroman_Γ.

To make this precise, for w∈T1(ℍ2)ݑ¤subscriptݑ‡1superscriptℍ2w\in T_1(\mathbbH^2)italic_w ∈ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), T,L>0ݑ‡ݐ¿0T,L>0italic_T , italic_L >0, let μ=μw,T,Lݜ‡subscriptݜ‡ݑ¤ݑ‡ݐ¿\mu=\mu_w,T,Litalic_μ = italic_μ start_POSTSUBSCRIPT italic_w , italic_T , italic_L end_POSTSUBSCRIPT be the probability measure on the set

0≤t≤Te-L/2conditional-setsubscriptݑ¤ݑ¡subscriptݐºݐ¿2subscriptݑ¤ݑ¡subscriptݐºݐ¿2subscriptݑ‡1superscriptℍ2subscriptݑ‡1superscriptℍ2 0ݑ¡ݑ‡superscriptݑ’ݐ¿2\displaystyle\big\{}(w_tG_-L/2,w_tG_L/2)\in T_1(\mathbbH^2)% \times T_1(\mathbbH^2)\,\big\,0\leq t\leq Te^-L/2\big{\} ( italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT - italic_L / 2 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_L / 2 end_POSTSUBSCRIPT ) ∈ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) × italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
induced by Lebesgue measure on [0,Te-L/2]0ݑ‡superscriptݑ’ݐ¿2[0,Te^-L/2][ 0 , italic_T italic_e start_POSTSUPERSCRIPT - italic_L / 2 end_POSTSUPERSCRIPT ]. Here wtsubscriptݑ¤ݑ¡w_titalic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is the unit vector obtained by pushing wݑ¤witalic_w to its right along an orthogonal geodesic for time tݑ¡titalic_t at unit speed. GLsubscriptݐºݐ¿G_Litalic_G start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT is the geodesic flow for time Lݐ¿Litalic_L. So if σtsubscriptݜŽݑ¡\sigma_titalic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is the segment of length Lݐ¿Litalic_L tangent to wtsubscriptݑ¤ݑ¡w_titalic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT with wtsubscriptݑ¤ݑ¡w_titalic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT based at its midpoint, then wtG-L/2subscriptݑ¤ݑ¡subscriptݐºݐ¿2w_tG_-L/2italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT - italic_L / 2 end_POSTSUBSCRIPT and wtGL/2subscriptݑ¤ݑ¡subscriptݐºݐ¿2w_tG_L/2italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_L / 2 end_POSTSUBSCRIPT are the unit vectors tangent to σtsubscriptݜŽݑ¡\sigma_titalic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT at its ends and oriented consistently with wtsubscriptݑ¤ݑ¡w_titalic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. Let π*(μw,T,L)subscriptݜ‹subscriptݜ‡ݑ¤ݑ‡ݐ¿\pi_{*}(\mu_w,T,L)italic_π start_POSTSUBSCRIPT * end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_w , italic_T , italic_L end_POSTSUBSCRIPT ) be the projection of μw,T,Lsubscriptݜ‡ݑ¤ݑ‡ݐ¿\mu_w,T,Litalic_μ start_POSTSUBSCRIPT italic_w , italic_T , italic_L end_POSTSUBSCRIPT to T1(S)×T1(S)subscriptݑ‡1ݑ†subscriptݑ‡1ݑ†T_1(S)\times T_1(S)italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) × italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ).

Theorem 1.4.

Assume Comm(Γ)0subscriptitalic-ϵ00\epsilon_0>0italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT >0 there exists a T0>0subscriptݑ‡00T_0>0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT >0 such that for all T,L>T0ݑ‡ݐ¿subscriptݑ‡0T,L>T_0italic_T , italic_L >italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and w∈T1(ℍ2)ݑ¤subscriptݑ‡1superscriptℍ2w\in T_1(\mathbbH^2)italic_w ∈ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )

|π*(μw,T,L)(f)-λ×λ(f)|<ϵ0.subscriptݜ‹subscriptݜ‡ݑ¤ݑ‡ݐ¿ݑ“ݜ†ݜ†ݑ“subscriptitalic-ϵ0\displaystyle|\pi_{*}(\mu_w,T,L)(f)-\lambda\times\lambda(f)|| italic_π start_POSTSUBSCRIPT * end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_w , italic_T , italic_L end_POSTSUBSCRIPT ) ( italic_f ) - italic_λ × italic_λ ( italic_f ) |
The condition Comm(Γ)
To give a sample of what can be obtained, for l1
Theorem 1.5.

Assume Comm(Γ)0subscriptsubscriptݜŽݐ¿ݐ¿0\\sigma_L\_L>0 italic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUBSCRIPT italic_L >0 end_POSTSUBSCRIPT be a sequence of oriented geodesic segments σL⊂SsubscriptݜŽݐ¿ݑ†\sigma_L\subset Sitalic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ⊂ italic_S. Assume length(σL)eL/2→∞normal-→ݑ™ݑ’ݑ›ݑ”ݑ¡ℎsubscriptݜŽݐ¿superscriptݑ’ݐ¿2length(\sigma_L)e^L/2\to\inftyitalic_l italic_e italic_n italic_g italic_t italic_h ( italic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT → ∞ as L→∞normal-→ݐ¿L\to\inftyitalic_L → ∞. Let (l1,l1),(a1,a2)subscriptݑ™1subscriptݑ™1subscriptݑŽ1subscriptݑŽ2(l_1,l_1),(a_1,a_2)( italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) be finite intervals of the real line. Let NL=NL(a1,a2,l1,l2)subscriptݑݐ¿subscriptݑݐ¿subscriptݑŽ1subscriptݑŽ2subscriptݑ™1subscriptݑ™2N_L=N_L(a_1,a_2,l_1,l_2)italic_N start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) be the set of unit vectors vݑ£vitalic_v such that

•
the basepoint of vݑ£vitalic_v is in σLsubscriptݜŽݐ¿\sigma_Litalic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT,

•
vݑ£vitalic_v is tangent to a closed geodesic γ∈ݒ¢L(l1,l2)ݛ¾subscriptݒ¢ݐ¿subscriptݑ™1subscriptݑ™2\gamma\in\cal G_L(l_1,l_2)italic_γ ∈ caligraphic_G start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ),

•
vݑ£vitalic_v is oriented consistently with γݛ¾\gammaitalic_γ,

•
the angle from vݑ£vitalic_v to σLsubscriptݜŽݐ¿\sigma_Litalic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT is in the interval π/2+(a1,a2)e-L/2ݜ‹2subscriptݑŽ1subscriptݑŽ2superscriptݑ’ݐ¿2\pi/2+(a_1,a_2)e^-L/2italic_π / 2 + ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_L / 2 end_POSTSUPERSCRIPT.

Then

#NL∼length(σL)vol(T1(S))(a2-a1)(el2-el1)eL/2.similar-to#subscriptݑݐ¿ݑ™ݑ’ݑ›ݑ”ݑ¡ℎsubscriptݜŽݐ¿ݑ£ݑœݑ™subscriptݑ‡1ݑ†subscriptݑŽ2subscriptݑŽ1superscriptݑ’subscriptݑ™2superscriptݑ’subscriptݑ™1superscriptݑ’ݐ¿2\displaystyle\#N_L\sim\fraclength(\sigma_L)vol(T_1(S))(a_2-a_1)(% e^l_2-e^l_1)e^L/2.# italic_N start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∼ divide start_ARG italic_l italic_e italic_n italic_g italic_t italic_h ( italic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) end_ARG start_ARG italic_v italic_o italic_l ( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) ) end_ARG ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_e start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT .

Here and throughout the paper, F∼Gsimilar-toݐ¹ݐºF\sim Gitalic_F ∼ italic_G means limL→∞FG=1subscript→ݐ¿ݐ¹ݐº1\lim_L\to\infty\fracFG=1roman_lim start_POSTSUBSCRIPT italic_L → ∞ end_POSTSUBSCRIPT divide start_ARG italic_F end_ARG start_ARG italic_G end_ARG = 1. vol(T1(S))ݑ£ݑœݑ™subscriptݑ‡1ݑ†vol(T_1(S))italic_v italic_o italic_l ( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) ) equals 2πarea(S)=(2π)2(2genus(S)-2)2ݜ‹ݑŽݑŸݑ’ݑŽݑ†superscript2ݜ‹22ݑ”ݑ’ݑ›ݑ¢ݑ ݑ†22\pi area(S)=(2\pi)^2(2genus(S)-2)2 italic_π italic_a italic_r italic_e italic_a ( italic_S ) = ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_g italic_e italic_n italic_u italic_s ( italic_S ) - 2 ). We also give a new proof of a special case of Rufus Bowen’s equidistribution theorem (see theorem 3.6).

1.2 Counting Pants Immersions

We use the theorems above to build and count pants immersions into Sݑ†Sitalic_S by constructing generators for the image subgroup corresponding to two of the boundary components. To state the results, fix ϵ>0italic-ϵ0\epsilon>0italic_ϵ >0. For r1,r2,r3,L>0subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3ݐ¿0r_1,r_2,r_3,L>0italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_L >0 let ݒ«L(r1,r2,r3)subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3\cal P_L(r_1,r_2,r_3)caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) be the set of all locally isometric orientation-preserving immersions j:P→S:ݑ—→ݑƒݑ†j:P\to Sitalic_j : italic_P → italic_S in which PݑƒPitalic_P is a hyperbolic three-holed sphere (i.e. a pair of pants) with geodesic boundary components of length l1,l2,l3subscriptݑ™1subscriptݑ™2subscriptݑ™3l_1,l_2,l_3italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT satisfying

li∈riL+(-ϵ,ϵ)subscriptݑ™ݑ–subscriptݑŸݑ–ݐ¿italic-ϵitalic-ϵ\displaystyle l_i\in r_iL+(-\epsilon,\epsilon)italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_L + ( - italic_ϵ , italic_ϵ )
for i=1,2,3ݑ–123i=1,2,3italic_i = 1 , 2 , 3. We implicitly identify immersions j1:P1→S:subscriptݑ—1→subscriptݑƒ1ݑ†j_1:P_1\to Sitalic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_S and j2:P2→S:subscriptݑ—2→subscriptݑƒ2ݑ†j_2:P_2\to Sitalic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → italic_S if there is an isometry Ψ:P1→P2:Ψ→subscriptݑƒ1subscriptݑƒ2\Psi:P_1\to P_2roman_Ψ : italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT such that j1=j2∘Ψsubscriptݑ—1subscriptݑ—2Ψj_1=j_2\circ\Psiitalic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ roman_Ψ. Thus ݒ«L(r1,r2,r3)subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3\cal P_L(r_1,r_2,r_3)caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) is a finite set. The first result gives asymptotics for the cardinality ݒ«L(r1,r2,r3)subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3\cal P_L(r_1,r_2,r_3)caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ).

Corollary 1.6.

Assume Comm(Γ)
|ݒ«L(r1,r2,r3)|∼8(eϵ/2-e-ϵ/2)3vol(T1(S))|Isom+(r1,r2,r3)|e(r1+r2+r3)L/2similar-tosubscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ38superscriptsuperscriptݑ’italic-ϵ2superscriptݑ’italic-ϵ23ݑ£ݑœݑ™subscriptݑ‡1ݑ†ݐ¼ݑ ݑœsuperscriptݑšsubscriptݑŸ1subscriptݑŸ2subscriptݑŸ3superscriptݑ’subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3ݐ¿2\displaystyle|\cal P_L(r_1,r_2,r_3)|\sim\frac8(e^\epsilon/2-e^-% \epsilon/2)^3vol(T_1(S))e^(r_1+r_2+r% _3)L/2| caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) | ∼ divide start_ARG 8 ( italic_e start_POSTSUPERSCRIPT italic_ϵ / 2 end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_ϵ / 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v italic_o italic_l ( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_S ) ) | italic_I italic_s italic_o italic_m start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) | end_ARG italic_e start_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_L / 2 end_POSTSUPERSCRIPT
where Isom+(r1,r2,r3)ݐ¼ݑ ݑœsuperscriptݑšsubscriptݑŸ1subscriptݑŸ2subscriptݑŸ3Isom^{+}(r_1,r_2,r_3)italic_I italic_s italic_o italic_m start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) is the orientation-preserving isometry group of the pair of pants with boundary lengths r1,r2,r3subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3r_1,r_2,r_3italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.

For comparison, recall that the number of closed oriented geodesics in Sݑ†Sitalic_S with length in (L-ϵ,L+ϵ)ݐ¿italic-ϵݐ¿italic-ϵ(L-\epsilon,L+\epsilon)( italic_L - italic_ϵ , italic_L + italic_ϵ ) is asymptotic to (eϵ-e-ϵ)eL/Lsuperscriptݑ’italic-ϵsuperscriptݑ’italic-ϵsuperscriptݑ’ݐ¿ݐ¿(e^\epsilon-e^-\epsilon)e^L/L( italic_e start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_ϵ end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT / italic_L (see e.g. [Buser]). Unlike |ݒ«L(r1,r2,r3)|subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3|\cal P_L(r_1,r_2,r_3)|| caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) | it does not depend on the genus of the surface.

We will prove this as a corollary to theorem 1.7 below. Recall that a closed oriented geodesic γݛ¾\gammaitalic_γ, is a local isometry γ:S1→S:ݛ¾→superscriptݑ†1ݑ†\gamma:S^1\to Sitalic_γ : italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → italic_S from the circle of length length(γ)ݑ™ݑ’ݑ›ݑ”ݑ¡ℎݛ¾length(\gamma)italic_l italic_e italic_n italic_g italic_t italic_h ( italic_γ ) to Sݑ†Sitalic_S. We identify geodesics γ1,γ2subscriptݛ¾1subscriptݛ¾2\gamma_1,\gamma_2italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if there is an orientation-preserving isometry Ψ:S1→S1:Ψ→superscriptݑ†1superscriptݑ†1\Psi:S^1\to S^1roman_Ψ : italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT such that γ1=γ2∘Ψsubscriptݛ¾1subscriptݛ¾2Ψ\gamma_1=\gamma_2\circ\Psiitalic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ roman_Ψ. Let γ¯¯ݛ¾\bar\gammaover¯ start_ARG italic_γ end_ARG denote the image of γݛ¾\gammaitalic_γ so length(γ¯)=length(γ)/mݑ™ݑ’ݑ›ݑ”ݑ¡ℎ¯ݛ¾ݑ™ݑ’ݑ›ݑ”ݑ¡ℎݛ¾ݑšlength(\bar\gamma)=length(\gamma)/mitalic_l italic_e italic_n italic_g italic_t italic_h ( over¯ start_ARG italic_γ end_ARG ) = italic_l italic_e italic_n italic_g italic_t italic_h ( italic_γ ) / italic_m where the map γ:S1→γ¯:ݛ¾→superscriptݑ†1¯ݛ¾\gamma:S^1\to\bar\gammaitalic_γ : italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → over¯ start_ARG italic_γ end_ARG is an mݑšmitalic_m-fold cover.

For γ∈ݒ¢r1L=ݒ¢r1L(-ϵ,ϵ)ݛ¾subscriptݒ¢subscriptݑŸ1ݐ¿subscriptݒ¢subscriptݑŸ1ݐ¿italic-ϵitalic-ϵ\gamma\in\cal G_r_1L=\cal G_r_1L(-\epsilon,\epsilon)italic_γ ∈ caligraphic_G start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = caligraphic_G start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( - italic_ϵ , italic_ϵ ), let ݒ«L(r1,r2,r3;γ)⊂ݒ«L(r1,r2,r3)subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3ݛ¾subscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3\cal P_L(r_1,r_2,r_3;\gamma)\subset\cal P_L(r_1,r_2,r_3)caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ; italic_γ ) ⊂ caligraphic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) denote the subset of immersions (j:P→S):ݑ—→ݑƒݑ†(j:P\to S)( italic_j : italic_P → italic_S ) in which jݑ—jitalic_j restricted to some boundary component is equivalent to γݛ¾\gammaitalic_γ.

Theorem 1.7.

Assume Comm(Γ)r1subscriptݑŸ2subscriptݑŸ3subscriptݑŸ1r_2+r_3>r_1italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT >italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r1+r3>r2subscriptݑŸ1subscriptݑŸ3subscriptݑŸ2r_1+r_3>r_2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT >italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT then

|ݒ«L(r1,r2,r3;γL)|∼4(eϵ/2-e-ϵ/2)2vol(T1(S))nlength(γ¯L)exp(-length(γL)/2+r2L/2+r3L/2)similar-tosubscriptݒ«ݐ¿subscriptݑŸ1subscriptݑŸ2subscriptݑŸ3subscriptݛ¾ݐ¿4superscriptsuperscriptݑ’italic-ϵ2superscriptݑ’italic-ϵ22ݑ£ݑœݑ™subscriptݑ‡1ݑ†ݑ›ݑ™ݑ’ݑ›ݑ”ݑ¡ℎsubscript¯ݛ¾ݐ¿ݑ™ݑ’ݑ›ݑ”ݑ¡ℎsubscri

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