FUCHSIAN GROUPS AND COMPACT HYPERBOLIC SURFACES

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1 FUCHSIAN GROUPS AND COMPACT HYPERBOLIC SURFACES YVES BENOIST AND HEE OH Abstract. We preset a topological proof of the followig theorem of Beoist-Quit: for a fiitely geerated o-elemetary discrete subgroup Γ 1 of PSL(2, R) with o parabolics, ad for a cocompact lattice Γ 2 of PSL(2, R), ay Γ 1 orbit o Γ 2\ PSL(2, R) is either fiite or dese. 1. Itroductio Let Γ 1 be a o-elemetary fiitely geerated discrete subgroup with o parabolic elemets of PSL(2, R). Let Γ 2 be a cocompact lattice i PSL(2, R). The followig is the first o-trivial case of a theorem of Beoist-Quit [1]. Theorem 1.1. Ay Γ 1 -orbit o Γ 2 \ PSL(2, R) is either fiite or dese. The proof of Beoist-Quit is quite ivolved eve i the case as simple as above ad i particular uses their classificatio of statioary measures [2]. The aim of this ote is to preset a short, ad rather elemetary proof. We will deduce Theorem 1.1 from the followig Theorem 1.2. Let H 1 = H 2 := PSL(2, R) ad G := H 1 H 2 ; H := {(h, h) : h PSL 2 (R)} ad Γ := Γ 1 Γ 2. Theorem 1.2. For ay x Γ\G, the orbit xh is either closed or dese. Our proof of Theorem 1.2 is purely topological, ad ispired by the recet work of McMulle, Mohammadi ad Oh [5] where the orbit closures of the PSL(2, R) actio o Γ 0 \ PSL(2, C) are classified for certai Kleiia subgroups Γ 0 of ifiite co-volume. While the proof of Theorem 1.2 follows closely the sectios 8-9 of [5], the argumets i this paper are simpler because of the assumptio that Γ 2 is cocompact. We remark that the approach of [5] ad hece of this paper is somewhat modeled after Margulis s origial proof of Oppeheim cojecture [4]. Whe Γ 1 is cocompact as well, Theorem 1.2 also follows from [6]. Fially we remark that accordig to [1], both Theorems 1.1 ad 1.2 are still true i presece of parabolic elemets, more precisely whe Γ 1 is ay o-elemetary discrete subgroup ad Γ 2 ay lattice i PSL(2, R). The topological method preseted here could also be exteded to this case. Oh was supported i part by NSF Grat. 1

2 2 YVES BENOIST AND HEE OH 2. Horocyclic flow o covex cocompact surfaces I this sectio we prove a few prelimiary facts about uipotet dyamics ivolvig oly oe factor H 1. The group PSL 2 (R) := SL 2 (R)/{±e} is the group of orietatio-preservig isometries of the hyperbolic plae H( 2 := {z ) C : Im z > 0}. The isometry a b correspodig to the elemet g = PSL c d 2 (R) is z az + b cz + d. It is implicit i this otatio that the matrices g stad for their equivalece class ±g i PSL 2 (R). This group PSL 2 (R) acts simply trasitively o the uit taget budle T 1 (H 2 ) ad we choose a idetificatio of PSL 2 (R) ad T 1 (H 2 ) so that the idetity elemet e correspods to the upward uit vector at i. We will also idetify the boudary of the hyperbolic plae with the exteded real lie H 2 = R { } which is topologically a circle. We recall that Γ 1 is a o-elemetary fiitely geerated discrete subgroup with o parabolic elemets of the group H 1 = PSL 2 (R), that is, Γ 1 is a covex cocompact subgroup. Let S 1 deote the hyperbolic orbifold Γ 1 \H 2, ad let Λ Γ1 H 2 be the limit set of Γ 1. Let A 1 ad U 1 be the subgroups of H 1 give by A 1 := {a t = ( ) e t/2 0 0 e t/2 : t R} ad U 1 := {u t = ( ) 1 t : t R}. 0 1 Let Ω Γ1 = {x Γ 1 \H 1 : xa 1 is bouded}. (2.1) As Γ 1 is a covex cocompact subgroup, Ω Γ1 is a compact A 1 -ivariat subset ad oe has the equality Ω Γ1 = {[h] Γ 1 \H 1 : h(0), h( ) Λ Γ1 }. I geometric words, see as a subset of the uit taget budle of S 1, the set Ω Γ1 is the uio of all the geodesic lies which stays iside the covex core of S 1. Defiitio 2.2. Let K > 1. A subset T R is called K-thick if, for ay t > 0, T meets [ Kt, t] [t, Kt]. Lemma 2.3. There exists K > 1 such that for ay x Ω Γ1, the subset T (x) := {t R : xu t Ω Γ1 } is K-thick. Proof. Usig a isometry, we may assume without loss of geerality that x = [e]. Sice the elemet e correspods to the upward uit vector at i, ad sice x belogs to Ω Γ1, both poits 0 ad belog to the limit set Λ Γ1. Sice u t ( ) = ad u t (0) = t, oe has the equality T (x) = {t R : t Λ Γ1 }. Write R Λ Γ1 as the uio J l where J l s are maximal ope itervals. Note that the miimum hyperbolic distace betwee the covex hulls i H 2 δ := if l m d(hull(j l), hull(j m ))

3 is positive, as 2δ is the legth of the shortest closed geodesic of the double of the covex core of S 1. Choose the costat K > 1 so that for t > 0, oe has d(hull[ Kt, t], hull[t, Kt]) = δ/2. Note that this choice of K is idepedet of t. If T (x) does ot itersect [ Kt, t] [t, Kt] for some t > 0, the the itervals [ Kt, t] ad [t, Kt] must be icluded i two distict itervals J l ad J m, sice 0 Λ Γ1. This cotradicts the choice of K. Lemma 2.4. Let K > 1 ad let T be a K-thick subset of R. For ay sequece h i H 1 U 1 covergig to e, there exists a sequece t T such that the sequece u t h u t has a limit poit i U 1 {e}. ( ) a b Proof. Write h =. We compute c d ( ) a c q := u t h u t = t (a d c t )t + b. c d + c t Sice the elemet h does ot belog to U 1, it follows that the (1, 2)-etries P (t ) := (a d c t )t +b are o-costat polyomial fuctios of t of degree at most 2 whose coefficiets coverge to 0. Hece, by Lemma 2.5 below, we ca choose t T goig to so that k P (t ) 1, for some costat k > 0 depedig oly o K. Sice the etry P (t ) is bouded ad sice h coverges to e, the product c t must coverge to 0 ad the sequece q has a limit poit i U 1 {e}. We have used the followig basic lemma : Lemma 2.5. For every K > 1 ad d 1, there exists k > 0 such that, for every o-costat polyomial P of degree d with P (0) k, ad for every K-thick subset T of R, there exists t i T such that k P (t) 1. Proof. Usig a suitable homothety i the variable t, we ca assume with o loss of geerality that P belogs to the set P d of polyomials of degree at most d such that P (1) = max P (t) = 1. [ 1,1] Assume by cotradictio that there exists a sequece P of polyomials i P d ad a sequece of K-thick subsets T of R such that P (t) sup T [ 1,1] coverge to 0. After extractio, the sequece T coverges to a K-thick subset T ad the sequece P coverges to a polyomial P P d which is equal to 0 o the set T [ 1, 1]. This is ot possible sice this set is ifiite. We record also, for further use, the followig classical lemma : Lemma 2.6. Let U 1 + be the semigroup {u t : t 0}. If the quotiet space X 1 := Γ 1 \H 1 is compact, ay U 1 + -orbit is dese i X 1. 3

4 4 YVES BENOIST AND HEE OH Proof. For x X 1, set x := xu. We the have x u U 1 + = xu 1 +. Hece if z is a limit poit of the sequece x i X 1, we have zu xu 1 +. By Hedlud s theorem [3], zu is dese. Hece the orbit xu 1 + is also dese. 3. Proof of Theorems 1.1 ad 1.2 I this sectio, usig miimal sets ad uipotet dyamics o the product space Γ\G, we provide a proof of Theorem Uipotet dyamics. We recall the otatio G := PSL 2 (R) PSL 2 (R) ad Γ := Γ 1 Γ 2. Set H 1 = {(h, e)}, H 2 = {(e, h)}, H = {(h, h)}; U 1 = {(u t, e)}, U 2 = {(e, u t )}, U = {(u t, u t )}; A 1 = {(a t, e)}, A 2 = {(e, a t )}, A = {(a t, a t )}; X 1 = Γ 1 \H 1, X 2 = Γ 2 \H 2, X = Γ\G = X 1 X 2. Recall that Γ 1 is a o-elemetary fiitely geerated discrete subgroup of H 1 with o parabolic elemets ad that Γ 2 is a cocompact lattice i H 2. For simplicity, we write ũ t for (u t, u t ) ad ã t for (a t, a t ). Note that the ormalizer of U i G is AU 1 U 2. Lemma 3.1. Let g be a sequece i G AU 1 U 2 covergig to e, ad let T be a K-thick subset of R for some K > 1. The for ay eighborhood G 0 of e i G, there exist sequeces s T ad t R such that the sequece ũ s g ũ t has a limit poit q e i AU 2 G 0. Proof. Fix 0 < ε 1. Write g = (g (1), g (2) the products q := ũ s g ũ t Set q (i) = u s g (i) u t = are give by ( a (i) c (i) s c (i) t = b(1) (b (i) d (1) s c (1) s a (1) ) with g (i) = ( a (i) c (i) b (i) d (i) ) d (i) s ) t (c (i) s a (i) d (i) + c (i) t The differeces q e are ow ratioal fuctios i s of the form q e = 1 c (1) s a (1). P (s ), ). The where P (s) is a polyomial fuctio of s of degree at most 2 with values i M 2 (R) M 2 (R). Sice the elemets g do ot belog to AU 1 U 2, these polyomials P are o-costats. I particular, the real valued polyomial fuctios s P (s) 2 are o-costat of degree at most 4. Sice P (0) 0 as, it follows from Lemma 2.5 that for ay 0 < ɛ, we ca choose s T goig to so that kε P (s ) ε for some costat k > 0 depedig oly o K. Moreover we ca deduce ).

5 1/2 c (1) s a (1) 2 from the coditio P (s ) ε by lookig at the (1, 1) ad (2, 2) etries of the first compoet of P (s ). Therefore kε/2 q e 2ε. By costructio, whe ε is small eough, the sequece q has a limit poit q e i A 1 A 2 U 2 G 0. We claim that this limit q = (q (1), q (2) ) belogs to the group AU 2. It suffices to check that the diagoal etries of q (1) ad q (2) are equal. If ot, the two sequeces c (i) s coverge to real umbers c (i) with c (1) c (2), ad a simple calculatio shows that the (1, 2)- etries of q (2) are comparable to c (2) c (1) c (1) 1 s which teds to, yieldig a cotradictio. Hece q belogs to AU H-miimal ad U-miimal subsets. Let Ω := Ω Γ1 X 2 where Ω Γ1 X 1 is defied i (2.1). Note that, sice Γ 2 is cocompact, oe has the equality Ω Γ2 = X 2. Let x = (x 1, x 2 ) Γ\G ad cosider the orbit xh. Note that xh itersects Ω o-trivially. Let Y be a H-miimal subset of the closure xh with respect to Ω, i.e., Y is a closed H-ivariat subset of xh such that Y Ω ad the orbit yh is dese i Y for ay y Y Ω. Sice ay H orbit itersects Ω, it follows that yh is dese i Y for ay y Y. Let Z be a U-miimal subset of Y with respect to Ω. Sice Ω is compact, such miimal sets Y ad Z exist. Set I the followig, we assume that Y = Y Ω ad Z = Z Ω. the orbit xh is ot closed ad aim to show that xh is dese i X. Lemma 3.2. For ay y Y, the idetity elemet e is a accumulatio poit of the set {g G H : yg xh}. Proof. If y does ot belog to xh, there exists a sequece h H such that xh coverges to y. Hece there exists a sequece g G covergig to e such that xh = yg. These elemets g do ot belog to H; hece provig the claim. Suppose ow that y belogs to xh. If the claim does ot hold, the for a sufficietly small eighborhood G 0 of e i G, the set yg 0 Y is icluded i the orbit yh. This implies that the orbit yh is a ope subset of Y. The miimality of Y implies that Y = yh, cotradictig the assumptio that the orbit yh = xh is ot closed. Lemma 3.3. There exists a elemet v U 2 {e} such that Zv xh. 5

6 6 YVES BENOIST AND HEE OH Proof. Choose a poit z = (z 1, z 2 ) Z. By Lemma 3.2, there exists a sequece g i G H covergig to e such that zg xh. We may assume without loss of geerality that g belogs to H 2. Suppose first that at least oe g belogs to U 2. Set v = g be oe of those belogig to U 2, so that the poit zv belogs to xh. Sice v commutes with U ad Z is U-miimal with respect to Ω, oe has the equality Zv = zvu, hece the set Zv is icluded i xh. Now suppose that g does ot belog to U 2. The, sice the set T (z 1 ) is K-thick for some K > 1 by Lemma 2.3, it follows from Lemma 2.4 that there exist a sequece t i T (z 1 ) such that, after extractio, the products ũ t g ũ t coverge to a elemet v U 2 {e}. Sice the poits zũ t belog to Ω, this sequece has a limit poit z Z. Sice oe has the equality z v = lim zũ t (ũ t g ũ t ) = lim (zg )ũ t, the poit z v belogs to xh. We coclude as i the first case that the set Zv = z vu is icluded i xh. Lemma 3.4. For ay z Z, there exists a sequece g i G U covergig to e such that zg Z for all. Proof. Sice the group Γ 2 is cocompact, it does ot cotai uipotet elemets ad hece the orbit zu is ot compact. By Lemma 2.3, the orbit zu is recurret i Z, hece the set Z zu cotais at least oe poit. Call it z. Sice the orbit z U is dese i Z, there exists a sequece ũ t U such that z = lim z ũ t. Hece oe ca write z ũ t = zg with g i G U covergig to e. Propositio 3.5. There exists a oe-parameter semi-group L + AU 2 such that ZL + Z. Proof. It suffices to fid, for ay eighborhood G 0 of e, a elemet q e i AU 2 G 0 such that the set Zq is icluded i Z; the writig q = exp w for a elemet w of the Lie algebra of G, we ca take L + to be the semigroup w w {exp(sw ) : s 0} where w is a limit poit of the elemets whe the diameter of G 0 shriks to 0. Fix a poit z = (z 1, z 2 ) Z. Accordig to Lemma 3.4 there exists a sequece g G U covergig to e such that zg Z. Suppose first that g belogs to AU 1 U 2 for ifiitely may ; the oe ca fid ũ t U such that the product q := g ũ t belogs to AU 2 {e} ad zq belogs to Z. Sice q ormalizes U ad sice Z is U-miimal with respect to Ω, oe has the equality Zq = zuq = zq U, hece the set Zq is icluded i Z. Now suppose that g is ot i AU 1 U 2. By Lemmas 2.3 ad 3.1, there exist sequeces s T (z 1 ) ad t R such that, after passig to a subsequece, the products ũ s g ũ t coverge to a elemet q e i AU 2 G 0. Sice

7 the elemets zũ s belog to Z, they have a limit poit z Z. Sice we have z q = lim zũ s (ũ s g ũ t ) = lim (zg )ũ t, the elemet z q belogs to Z. We coclude as i the first case that the set Zq = z qu is icluded i Z. Propositio 3.6. There exist a elemet z xh ad a oe-parameter semi-group U + 2 U 2 such that zu + 2 xh. Proof. By Propositio 3.5 there exists a oe-parameter semigroup L + AU 2 such that ZL + Z. This semigroup L + is equal to oe of the followig: U 2 +, A+ or v0 1 A+ v 0 for some elemet v 0 U 2 {e}, where U 2 + ad A+ are oe-parameter semigroups of U 2 ad A respectively. Whe L + = U 2 +, our claim is proved. Suppose ow L + = A +. By Lemma 3.3 there exists a elemet v U 2 {e} such that Zv xh. The oe has the iclusios ZA + va ZvA xha xh. Choose a poit z Z ad a sequece ã t A + goig to. Sice z ã t belog to Ω, after passig to a subsequece, the sequece z ã t coverges to a poit z xh Ω. Moreover, sice the Hausdorff limit of the sets ã t A + is A, oe has the iclusios zava lim z ã t (ã t A + )va = z A + va xh. Now by a simple computatio, we ca check that the set AvA cotais a oe-parameter semigroup U 2 + of U 2, ad hece the orbit zu 2 + is icluded i xh as desired. Suppose fially L + = v0 1 A+ v 0 for some v 0 i U 2 {e}. We ca write A + = {ã εt : t 0} with ε = ±1 ad v 0 = (e, u s ) with s 0. A simple computatio shows that the set U 2 := {(e, u εst) : 0 t 1} is icluded i v0 1 A+ v 0 A. Hece oe has the iclusios ZU 2 Zv 1 0 A+ v 0 A ZA xh. Choose a poit z Z ad let z xh be a limit of a sequece z ã t with t goig to +. Sice the Hausdorff limit of the sets ã t U 2ã t is the semigroup U 2 + := {(e, u εst) : t 0}, oe has the iclusios zu Coclusio. lim (z ã t )ã t U 2ã t ZU 2 A xh. Proof of Theorem 1.2. Suppose that the orbit xh is ot closed. By Propositio 3.6, the orbit closure xh cotais a orbit zu 2 + of a oe-parameter subsemigroup of U 2. Sice Γ 2 is cocompact i H 2, by Lemma 2.6, this orbit zu 2 + is dese i zh 2. Hece we have the iclusios X = zg = zh 2 H zu + 2 H xh. 7

8 8 YVES BENOIST AND HEE OH This proves the claim. Proof of Theorem 1.1. Let x = [g] be a poit of X 2 = Γ 2 \H 2. By replacig Γ 1 by g 1 Γ 1 g, we may assume without loss of geerality that g = e. Oe deduces Theorem 1.1 from Theorem 1.2 thaks to the followig equivaleces: The orbit [e]h is closed (resp. dese) i Γ\G The orbit Γ[e] is closed (resp. dese) i G/H The product Γ 2 Γ 1 is closed (resp. dese) i PSL 2 (R) The orbit [e]γ 1 is closed (resp. dese) i Γ 2 \ PSL 2 (R). Refereces [1] Y. Beoist ad J. F. Quit Statioary measures ad ivariat subsets of homogeeous spaces I. Aals of Math, Vol 174 (2011), p [2] Y. Beoist ad J. F. Quit Statioary measures ad ivariat subsets of homogeeous spaces III. Aals of Math, Vol 178 (2013), p [3] G. Hedlud Fuchsia groups ad trasitive horocycles. Duke Math. J. Vol 2, 1936, p [4] G. Margulis Idefiite quadratic forms ad uipotet flows o homogeeous spaces. I Dyamical systems ad ergodic theory (Warsaw, 1986), volume 23. Baach Ceter Publ., [5] C. McMulle, A. Mohammadi ad H. Oh Geodesic plaes i hyperbolic 3-maifolds. Preprit, 2015 [6] M. Rater Raghuatha s topological cojecture ad distributios of uipotet flows. Duke Math. J. 63 (1991), p Uiversite Paris-Sud, Batimet 425, Orsay, Frace address: yves.beoist@math.u-psud.fr Mathematics departmet, Yale uiversity, New Have, CT ad Korea Istitute for Advaced Study, Seoul, Korea address: hee.oh@yale.edu