Some Examples of Limits of Kleinian Groups

Transcription

1 Some Examples of Limits of Kleiia Groups Jeff Brock August 21, 2007 The goal of this lecture will be to exhibit the thick-thi decompositio for a collectio of examples of Kleiia groups. I this lecture, we ll demostrate how geometric limits, as opposed to algebraic limits, ca be a good tool to do this. The limitig pheomea that we would like to i this lecture discuss are the followig: 1. (Jørgese) A sequece of represetatio ρ : Z Isom + (H 3 ), where {ρ (Z)} coverges geometrically to a group isomorphic to Z Z 2. (Kerchoff-Thursto) A sequece of quasi-fuchsia groups give by iteratio of a Deh twist o a Bers slice, such that Q(X, T X) has o parabolics but coverges geometrically to a maifold with a rak two cusp. 3.(Jørgese-Thursto-Boaho-Otal) A sequece of groups Q that coverges geometrically to group whose quotiet maifold has ifiitely may cusps. This is doe by doig higher ad higher Deh twists o a sequece of curves. I each of these examples, the boudaries of the thi parts remai cotrolled, so the thick part of the maifold stabilizes. I the ext example this is ot the case. 4. A sequece of groups Q(X, φ X), where φ is a partially pseudo-aosov mappig class, i.e. whe φ restricted to some compoet or compoets is a pseudo-aosov, ad whe restricted to others is the idetity map. β α As a example of this last type, suppose φ were give by τ 1 α τ β, where α ad β are the 1

2 curves show above. The φ is a pseudo-aosov whe restricted to the oe-holed torus o the right of the above diagram. I this case, the covex core boudaries o oe side of the maifold stay close together, as there is a figure eight homotopy betwee them. To push curves through the other side of the maifold, however, takes a very log time, ad we see a large Margulis tube appearig. Short homotopy Margulis tube Schematic Picture Because the restrictio of φ to the right had side of the surface is pseudo-aosov, the right half of the maifold looks icreasigly like a cyclic cover of a maifold fiberig over the circle with puctured torus fiber. I the approximates, we therefore see a maifold whose geometric structure o oe side is roughly periodic. 2

3 Jørgeses example- As a first example, [ we retur to] the sequece of represetatios e ω sih(ω ρ : Z PSL 2 (C) defied by ρ (1) = ), where ω 0 e ω = 1 + πi 2. To uderstad the geometric limit of these represetatios, we fix attetio o a sigle poit x. For ay, the loxodromic elemet ρ (1) correspods to twistig x aroud a coe whose axis is the axis of ρ (1). As icreases, the axis of ρ (1) moves further ad further from x. The followig illustratio shows these coes for m >. ρ ()x ρ (m)x m ρ (2)x ρ (1)x x x ρ (1) ad ρ () traslate x by roughly the same amout, but i differet directios alog the coe. As icreases, the coe through x becomes very flat, ad i the limit we see a horosphere through x. This is clearly visible i the ball model, i which the regular eigborhood of the loxodromic axis is a baaa rather tha a coe, ad oe sees the edpoits of its axis comig closer ad closer together. x By the defiitio of the geometric limit, both lim ρ (1) ad lim ρ () must be i the geometric limit, so the geometric limit has two geerators. α(x) β(x) α=lim ρ (1) g x β=lim ρ () g Note that lim ρ () is ot i the algebraic limit, as the algebraic limit cosists of limits 3

4 of represetatios of fixed elemets. The algebraic limit is geerated by the limits of the geerators, so i this case is a cyclic group geerated by the sigle parabolic lim ρ (1). While the algebraic limit is isomorphic to the origial group, ad the geometric limit is ot, though the quotiet maifold of the geometric limit looks much more like the approximates tha the quotiet maifold of the algebraic limit. The Kerhoff-Thursto Example- I this example, we look at a sequece of quasi-fuchsia groups whose covex core boudaries have cotrolled geometry, but whose geometric limit develops a rak-two cusp i its iterior. Let X be a poit i the Teichmüller space of geus two surfaces, ad τ the elemet of the modular group give by Deh twistig aroud the curve show below. Cosider the sequece of quasi-fuchsia three maifolds Q(X, τ X) give by simultaeous uiformizatio of X ad τ X. This sequece sits i the Bers slice based at X, which is precompact i the algebraic topology. As a result, by passig to a subsequece we ca esure that Q(X, τ X) coverges to Q = H 3 /Γ A. 1 Let Γ G be the geometric limit of the groups Q(X, τ X), M G the quotiet maifold. Theorem: (Kerckhoff-Thursto) M G = S R 0 Missig Curve Oe sees a rak two cusp appear i the limit comig from a tubular eighborhood of the curve. π 1 (N ()) = Z Z, ad this gives a rak two parabolic subgroup of the geometric limit. 1 Actually, this sequece coverges without passig to subsequece. 4

5 To prove this theorem, we fix a set of geerators for the fudametal group of the surface ad cosider how τ acts o these geerators. We choose a basepoit o oe side of the curve, ad geerators as show below. α 1 α 2 τ α 1 1 α 2 β β 2 1 β 1 β 2 1 As a automorphism of the fudametal group, τ fixes α 1 ad β 1, but cojugates α 2 ad β 2 by. As Q(X, τ X) Q = H 3 /Γ A, we ca choose represetatios ρ : π 1 (S) Isom + (H 3 ) with Q(X, τ X) = H 3 /ρ (π 1 (S)) ad ρ (π 1 (S)) Γ A as goes to ifiity. A key trick i aalyzig limits of such iteratios is to otice that by precomposig the represetatio by a automorphism of the group, oe does ot chage the image Kleiia group. If ρ = ρ τ 1, the the images of ρ ad ρ are the same quasi-fuchsia group. This sequece is give by remarkig the iitial surface usig τ, so the correspodig maifold is Q(τ X, τ τ X) = Q(τ X, X). The Bers slice {Q(Y, X) Y Teich(S)} is also precompact, so agai we ca fid a sequece ρ that coverges algebraically. I chagig from the sequece ρ to the sequece ρ, we have shifted our attetio from oe side of the quasi-fuchsia maifold to the other, however as far as the images of these represetatios are cocered we have doe othig. We are lookig at the same group, we have simply chaged our perspective o which ed is twisted. As the sequece of groups is the same, their geometric limits are the same group Γ G. We already have a list of elemets that must be i the group Γ G, amely the limits of ρ () ρ (α 1 ) = ρ (α 1 ), ρ (β 1 ) = ρ (β 1 ) ρ (α 2 ), ρ (β 2 ) ρ (α 2 ) = ρ ( )ρ (α 2 )ρ ( ), ρ (β 2 ) = ρ ( )ρ (β 2 )ρ ( ) We claim that ρ ( ) also coverges to some Isom(H 3 ). To see this, otice that ρ ( ) seds the fixed poits of ρ (α 2 ) to the fixed poits of ρ (α 2 ), as if ρ (α 2 )x = x, the ρ (α 2 )(ρ ( )x) = (ρ (α 2 )ρ ( ))x = (ρ ( )ρ (α 2 ))x = ρ ( )x. I fact, the same argumet shows that for ay word w α 2, β 2, ρ ( ) seds the set of fixed poits of ρ (w) to the set of fixed poits for ρ (w). As α 2 ad β 2 geerate a rak-2 free group, there are ifiitely may boudary poits that are fixed by words w α 2, β 2. We ca therefore pick three words w 1, w 2 ad w 3 ad distict poits o the sphere at ifiity x 1, x 2 ad x 3 such that ρ (w 1 )x 1 = x 1, ρ (w 2 )x 2 = x 2 ad ρ (w 3 )x 3 = x 3. As goes to ifiity, 5

6 x i coverges to a fixed poit x i of ρ (w i ) ad ρ ( )x i coverges to a fixed poit y i of ρ (w i ). Thus lim ρ ( )x i = y i. This shows that ρ ( ) coverges to the uique Möbius trasformatio that seds x 1 y 1, x 2 y 2 ad x 3 y 3. As ρ ( ), is i Γ G by the defiitio of the geometric limit. We claim that is ot i the algebraic limit. Suppose it were, i.e. suppose there exists a elemet g such that lim ρ (g) =. The ρ (g) ρ ( ) = ρ (g ) id. By Gromov s compactess theorem (see exercises), the geometric limit is a discrete group, so there is some lower boud ɛ o the ijectivity radius of all elemets of Γ G. Thus if ɛ is the ijectivity radius of ρ (π 1 (S)), the ρ (π 1 (S)) Γ G implies that ɛ ɛ. I particular, for large eough, the ijectivity radius of elemets i ρ (π 1 (S)) is bouded below by ɛ/2. As ρ (g ) id, for sufficietly large ρ (g ) has ijectivity radius below ɛ/2, ad therefore equals the idetity. The represetatios ρ are faithful, however, so this is a cotradictio. We ca also show, ρ () = Z Z. The first thig to check is that these two elemets commute. ρ ( ) ad ρ () commute for all, so the limit of their commutator is the idetity. This shows that, ρ () = Z Z or, ρ () = Z. If the latter is the case, the i = ρ () j for some i ad j, so ρ ( i )ρ ( j ) = ρ ( i j ) id. Applyig the same argumet as before, we have that ρ ( i j ) is equal to the idetity for sufficietly large, which agai cotradicts the faithfuless of the represetatio. We ow kow that we have a rak two free abelia subgroup, which must correspod to a cusp i a hyperbolic maifold. Because is homotopic to the core curve of this cusp i the approximates, the same is true i the limit, so the quotiet of H 3 by the group geerated by the elemets we have foud so far is homeomorphic to S R {0}. Oe ca use the theory of Klei-Maskit combiatio to show that there are o other elemets i the geometric limit, but we wo t give the details of that here. A Example where the Geometric Limit is ot Fiitely Geerated- Let η ad be a pair of fillig curves, i.e. homeomorphic to a disjoit uio of disks. curves η ad such that S {η } is η The previous example dealt with simultaeous uiformizatio of a surface ad a re-markig of that surface by a Deh twist about a sigle curve. I this example, we will Deh twist 6

7 about a sequece of curves. Cosider the sequece of maifolds Q(X, τ τ η X), Q(X, τ 2 τ η 2 τ 2 τ η 2 X),..., Q(X, τ τ η τ τ η... τ τ η X),... }{{} These have a algebraic limit Q by the compactess of the closure of the Bers slice, but this algebraic limit gives very little iformatio about the approximates. As far as the algebraic limit is cocered oly the fial t is importat. This is because the image of ay curve uder the mappig class τ τ η τ τ η... τ τ η will look like a curve that is highly twisted about. The algebraic limit of this sequece will look like the algebraic limit i the Kerckhoff-Thursto example, so we see a cusp i the limit correspodig to. By performig the same remarkig trick as i the previous example, however, we ca brig differet twists i the sequece ito focus, e.g. the algebraic limit of the sequece Q(τ 1 X, τ η X), Q(τ 2 X, τη 2 τ 2 τ η 2 X),..., Q(τ 1 X, τη τ τ η... τ τ η X),... }{{} will have a cusp correspodig to a differet curve. Iteratig this remarkig trick, we ca see a whole sequece of cusps. Just like i the previous example, the topology of our surfaces is S R after fillig i some deleted curves, however ow this sequece of curves goes off to ifiity.... Limits with missig subsurfaces- We will ow see what happes whe we iterate a reducible mappig class. Let φ be a mappig class o a geus two surface whose restrictio to oe half of the surface is a psuedo-aosov ad whose restrictio to the other is the idetity. The sequece of maifolds Q(X, φ (X) coverges algebraically to a group Q that is partially degeerate, where some of the eds have short curves exitig them ad others do t. If we look at Q(X, φ (X)) for large, we see a bouded homotopy o oe side of the maifold, as the curves α 1 ad β 1 o oe covex core boudary are homotopic to their represetatives o the other side of the covex core. Ay homotopy betwee curves o the pseudo-aosov side of the maifold, however, takes loger ad loger to occur. Oe way to see this is to otice that the bouded legth curves o oe side of the maifold are α 2 ad β 2, ad the bouded legth curves o the other side are φ (α 2 ) ad φ (β 2 ). As φ (α 2 ) 7

8 ad φ (β 2 ) itersect α 2 ad β 2 a lot, the collar lemma gives that α 2 ad β 2 must be very log o the side where φ (α 2 ) ad φ (β 2 ) have bouded legth. The geometric limit of this sequece is homeomorphic to S R S, where, S is the subsurface o which φ acts as a pseudo-aosov. Geometrically what we see i the approximates is a Margulis tube appearig which gives oe half of the maifold room to spread out, as was described at the begiig of this talk. This Margulis tube is differet from those i the previous examples, because its boudary does ot have cotrolled geometry. For large, the boudary of the thi part of the maifold becomes very large. Because φ restricted to S is a pseudo-aosov, the side of the maifold correspodig to S looks more ad more like the Z cover of a maifold fiberig over the circle with fiber S. The limit therefore has a periodic geometric structure o this side. A more faithful geometric picture of this maifold is show below. As the above picture shows, the geometric limit has ew degeerate eds, which hits at the fact that geometric limits of quasi-fuchsia groups ca be quite geeral. 8