On horospheric limit sets of Kleinian groups
aa r X i v : . [ m a t h . C V ] J un ON HOROSPHERIC LIMIT SETS OF KLEINIAN GROUPS
KURT FALK AND KATSUHIKO MATSUZAKI
Abstract.
In this paper we partially answer a question of P. Tukia about thesize of the difference between the big horospheric limit set and the horosphericlimit set of a Kleinian group. We mainly investigate the case of normal sub-groups of Kleinian groups of divergence type and show that this difference isof zero conformal measure by using another result obtained here: the Myrberglimit set of a non-elementary Kleinian group is contained in the horosphericlimit set of any non-trivial normal subgroup. Introduction and statement of results
In [29] Sullivan showed that the conservative part of the action of a Kleiniangroup G on its limit set coincides up to zero sets of the spherical Lebesgue measurewith the horospheric limit set L h ( G ) of G , i.e. the set of all limit points at whichevery horoball contains infinitely many orbit points of the group. Later, Tukia [36]generalised this result by showing that the same conservative part of the groupaction on the limit set coincides with the so-called big horospheric limit set L H ( G )up to zero sets of any conformal measure of dimension δ ( G ) for G . Here, δ ( G )denotes as usual the critical exponent of G and L H ( G ) consists of all limit pointsof G at which there exists a horoball containing infinitely many orbit points of G .For a generalisation of these observations to boundary actions of discrete groups ofisometries of Gromov hyperbolic spaces we refer the reader to Kaimanovich’s work[15].All this considered, Tukia [36] asked the very natural question of how big thedifference L H ( G ) \ L h ( G ) might be, also in light of the close relationship betweenso-called Garnett points [29] and this difference of sets (see [36] for more details).One possible first attempt at answering this question could make use of a strati-fication of the limit set of a Kleinian group between the radial and the horosphericlimit sets in terms of linear escape rates to infinity within the convex core of thecorresponding hyperbolic manifold. These ideas have been used by different au-thors in various slightly different guises and we refer to Section 3 for details. Sinceboth the radial limit set and the big horospheric limit set appear as elements of thisstratification, it would seem that it can be used to measure the difference betweenthe big horospheric and the horospheric limit set. However, Proposition 3.3 goesto show that we cannot detect this difference just by changing linear escape rates.In this paper we follow a different and somewhat surprising idea in order toanswer Tukia’s question in the case of normal subgroups of Kleinian groups of Mathematics Subject Classification.
Primary 30F40, Secondary 37F35.
Key words and phrases.
Kleinian group, horospheric limit set, Myrberg limit set, critical ex-ponent, Patterson measure, geodesic flow, Hausdorff dimension.The authors were supported by JSPS Grant-in-Aid for Challenging Exploratory Research divergence type. We show (see Theorem 5.2) that if N is some non-trivial normalsubgroup of the Kleinian group G of divergence type, then L H ( N ) \ L h ( N ) is anullset w.r.t. the uniquely determined conformal measure µ of dimension δ ( G ) for G . This, of course, is also a conformal measure of dimension δ ( G ) for N . Weobtain this result as a consequence of another observation (see Theorem 5.1) ofindependent interest, namely, that the Myrberg limit set of a Kleinian group G isalways contained in the horosperic limit set of any non-trivial normal subgroup N of G . This in turn is a refinement of the fact proven in [17] that in this situation theradial limit set of G is contained in the big horospheric limit set of N . Of course,one then also needs G to be of divergence type in order to ensure that the Myrberglimit set is of full µ -measure (see e.g. [35], [27] and [8] for more details). Thesurprising aspect of this approach is that it is an instance where a statement aboutan essentially non-conservative phenomenon (the difference between the horosphericand big horospheric limit set) is proven using a consequence of ergodicity. TheMyrberg limit set can be understood as a qualitative description of the ergodicityof the geodesic flow in the case of divergence type groups. We discuss these notionsin Section 2 and the beginning of Section 5.Clearly, one wants to measure the difference between the horospheric and bighorospheric limit set also in a more general case than for normal subgroups ofKleinian groups of divergence type. In view of our previous work [10], we conjecturethat the statement of Theorem 5.2 also holds for Kleinian groups for which theconvex hull of the limit set admits a uniformly distributed set whose Poincar´eseries diverges at its critical exponent (see the end of Section 5 and Conjecture 1for more details).Having answered Tukia’s question for normal subgroups N of groups G of diver-gence type by considering the Myrberg limit set of G , leaves the question open whyone should measure the size of the difference L H ( N ) \ L h ( N ) by a δ ( G )-dimensionalconformal measure, as we do, and not a δ ( N )-dimensional one. Recall that δ ( N )may very well be strictly smaller than δ ( G ). The answer, at least in our con-text, is given in Proposition 6.1 where we show that the Hausdorff dimension of L M ( G ) is equal to δ ( G ), provided G is of divergence type and the strong sublineargrowth limit set Λ ∗ ( G ) is of full measure w.r.t. some Patterson measure of G (seeSection 3 and Section 6 for more details). Already Sullivan [28] conjectured thatΛ ∗ ( G ) should be of full Patterson measure for all groups G of divergence type, butwe go one step further and conjecture that the Hausdorff dimension of the Myr-berg limit set coincides with the critical exponent for all non-elementary Kleiniangroups (Conjecture 2 in Section 6). One may be able to prove this by refining awell-known argument of Bishop and Jones [5] showing that the Hausdorff dimensionof the radial limit set is equal to the critical exponent for non-elementary groups.2. Preliminaries
Limit sets of a Kleinian group.
Let ( B n +1 , d ), n ≥
1, be the unit ball modelof ( n + 1)-dimensional hyperbolic space with the hyperbolic distance d . The n -dimensional unit sphere S n is the boundary at infinity of hyperbolic space. Kleiniangroups are discrete subgroups of the group of orientation preserving isometries ofhyperbolic space. The quotient M G = B n +1 /G of ( n + 1)-dimensional hyperbolicspace through a torsion free Kleinian group, that is, a group without elliptic ele-ments, is an ( n + 1)-dimensional hyperbolic manifold. OROSPHERIC LIMIT SETS 3
The limit set L ( G ) of a Kleinian group G is the set of accumulation points of anarbitrary G -orbit, and is a closed subset of S n . If L ( G ) consists of more than twopoints, then it is uncountable and perfect, and G is called non-elementary . Thehyperbolic convex hull of the union of all geodesics both of whose end points are in L ( G ) is called the convex hull of L ( G ), and is denoted by H ( L ( G )). The quotient C ( M G ) := H ( L ( G )) /G is called the convex core of M G . Equivalently, the convexcore is the smallest convex subset of M G containing all closed geodesics of M G .A non-elementary Kleinian group G is called convex cocompact if the convex core C ( M G ) is compact, and geometrically finite if some ε -neighbourhood of C ( M G ) hasfinite hyperbolic volume.A point ξ ∈ L ( G ) is a radial limit point of G if for any x ∈ B n +1 and for anygeodesic ray towards ξ there is a constant c ≥ Gx are within distance c of the given geodesic ray. The set of all radiallimit points of G is called the radial limit set and is denoted by L r ( G ).A point ξ ∈ L ( G ) is a horospheric limit point if for any x ∈ B n +1 every horospheretangent to S n at ξ contains an orbit point gx for some g ∈ G . The set of allhorospheric limit points of G is called the horospheric limit set and is denoted by L h ( G ). A point ξ ∈ L ( G ) is an element of the big horospheric limit set , denoted L H ( G ), if for x ∈ B n +1 there is some horosphere tangent to S n at ξ that containsinfinitely many orbit points in Gx . By definition, we have L h ( G ) ⊂ L H ( G ).2.2. The critical exponent and invariant conformal measures.
For a Kleiniangroup G and points x, z ∈ B n +1 , the Poincar´e series with exponent s > P s ( Gx, z ) := X g ∈ G e − s d ( g ( x ) ,z ) . The critical exponent δ = δ ( G ) of G is δ ( G ) := inf { s > | P s ( Gx, z ) < ∞} = lim sup R →∞ log B ( z, R ) ∩ Gx ) R , where B ( z, R ) is the hyperbolic ball of radius R centred at z and · ) denotes thecardinality of a set. By the triangle inequality, δ does not depend on the choiceof x, z ∈ B n +1 . If G is non-elementary, then 0 < δ ≤ n . Also, Roblin [26] showedthat the above upper limit is in fact a limit. G is called of convergence type if P δ ( Gx, z ) < ∞ , and of divergence type otherwise. It is known that a geometricallyfinite Kleinian group is of divergence type.A family of positive finite Borel measures { µ z } z ∈ B n +1 on S n is called s -conformalmeasure for s > { µ z } are absolutely continuous to each other and, for each z ∈ B n +1 and for almost every ξ ∈ S n , dµ z dµ o ( ξ ) = | g z ( ξ ) | s , where o is the origin in B n +1 , g z is a conformal automorphism of B n +1 sending z to o and | · | denotes the linear stretching factor of a conformal map. For a Kleiniangroup G , if { µ z } satisfies g ∗ µ g ( z ) = µ z (a . e) for every z ∈ B n +1 and for every g ∈ G ,then { µ z } is called G -invariant . KURT FALK AND KATSUHIKO MATSUZAKI
The measure µ = µ o can represent the family { µ z } and the G -invariance propertycan be rephrased as follows: for every g ∈ G and any measurable A ⊂ S n we have µ ( g ( A )) = Z A | g ′ ( ξ ) | s dµ ( ξ ) . If a positive finite Borel measure µ on S n satisfies this condition, we also call µ itself a G -invariant conformal measure of dimension s .We consider a G -invariant conformal measure of dimension δ = δ ( G ) supportedon the limit set L ( G ). The canonical construction of such a measure due to Pat-terson [21], [22] is as follows (See also [20]). Assume first that G is of divergencetype. For any s > δ , take a weighted sum of Dirac measures on the orbit Gx forsome x ∈ B n +1 : µ s ( x ) := 1 P s ( Gx, o ) X g ∈ G e − sd ( gx,o ) gx . We can choose some sequence s n > δ tending to δ such that µ s n ( x ) converges tosome measure µ on B n +1 in the weak- ∗ sense. Then we see that µ is a G -invariantconformal measure of dimension δ supported on L ( G ), which is called a Pattersonmeasure for G . When G is of convergence type, we need to use a modified Poincar´eseries e P s ( Gx, o ) to make it divergent at δ and apply a similar argument. If G is ofdivergence type, then a G -invariant conformal measure of dimension δ is unique upto multiplication by a positive constant, hence it is the Patterson measure.There is another canonical construction of G -invariant conformal measures, main-ly in the case where G is of convergence type. This was introduced briefly by Sul-livan in [32] and developed further in [3] (see also [11]). Suppose that the Poincar´eseries for G converges at dimension s ≥ δ . We again consider the weighted sum ofDirac measures µ s ( x ) as above, but here we move the orbit point x to some point ξ ∈ S n at infinity within a Dirichlet fundamental domain for G . We can choose asequence x n ∈ B n +1 tending to ξ such that µ s ( x n ) converges to some measure µ on B n +1 in the weak- ∗ sense. Then we see that µ is a G -invariant conformal measureof dimension s on S n , which is called an ending measure .2.3. Ergodicity of the geodesic flow.
For a hyperbolic manifold M G = B n +1 /G ,the unit tangent bundle T M G = F p ∈ M G T p M G is the union of the unit tangentvectors v ∈ T p M G at p taken over all p ∈ M G . Each element of T M G is representedby the pair ( v, p ). Let ˜ g ξ,z ( t ) be a geodesic line of unit speed in B n +1 starting froma given point z ∈ B n +1 towards ξ ∈ S n as t → ∞ . The unit tangent bundle T B n +1 of hyperbolic space is also represented by S n × B n +1 = { ( ξ, z ) } throughthe correspondence of the unit tangent vector d ˜ g ξ,z dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ˜ g ′ ξ,z (0)to ( ξ, z ). For ( v, p ) ∈ T M G , let g v,p ( t ) denote the geodesic line such that g v,p (0) = p and g ′ v,p (0) = v . We can assume that this is the projection of some geodesic line˜ g ξ,z ( t ) under B n +1 → M G . In this case, we also use the notation g ξ,z instead of g v,p .The geodesic flow φ t : T M G → T M G is a map sending ( v, p ) to ( g ′ v,p ( t ) , g v,p ( t ))for each t ∈ R .Any conformal measure µ on S induces a measure ˜ µ ∗ on the unit tangent bundle T B n +1 = S n × B n +1 that is invariant under the geodesic flow (Sullivan [28], see also OROSPHERIC LIMIT SETS 5 [20]). The unit tangent bundle T M G of the hyperbolic manifold M G is nothingbut the quotient of T B n +1 by the canonical action of G . If µ is invariant under G ,then so is ˜ µ ∗ and hence it descends to a measure µ ∗ on T M G .We say that the geodesic flow φ t is ergodic with respect to µ ∗ if for any measur-able subset E of T M G that is invariant under φ t for all t ∈ R we have that either µ ∗ ( E ) = 0 or µ ∗ ( T M G \ E ) = 0. Sullivan [28] (and Aaronson and Sullivan [1])generalised the result of Hopf [13], [14] to show the following (see also [25]). Theorem 2.1.
Let G be a Kleinian group and µ a G -invariant conformal measureof dimension δ ( G ) . Then the following conditions are equivalent: (i) µ ( L r ( G )) = µ ( S n ) ; (ii) the geodesic flow φ t is ergodic with respect to µ ∗ ; (iii) G is of divergence type. If G is geometrically finite, then the measure µ ∗ corresponding to the Pattersonmeasure µ is finite. If µ ∗ is a finite measure, then the geodesic flow φ t is ergodic withrespect to µ ∗ and hence all conditions from Theorem 2.1 hold ([31]). However, thereare also large classes of geometrically infinite groups for which µ ∗ is infinite andthese conditions are true ([33], [30], [24] or [1]). Moreover, there are also examplesof geometrically infinite groups for which µ ∗ is a finite measure and the conditionsfrom the theorem hold true ([23]).3. Limit sets between radial and horospheric
For a Kleinian group G , let µ be a G -invariant conformal measure on S n and X ⊂ S n a measurable subset that is invariant under G . The action of G is called conservative on X with respect to µ if any measurable subset A ⊂ X with µ ( A ) > µ ( A ∩ g ( A )) > g ∈ G . For the n -dimensional sphericalmeasure µ , Sullivan [29] showed that G acts conservatively on the horospheric limitset L h ( G ), and that the difference between L h ( G ) and L H ( G ) is actually of nullmeasure. Later, a characterization of the conservative action for a G -invariantconformal measure µ in general was obtained by Tukia [36]. In particular, if a G -invariant conformal measure µ has no point mass, then the conservative part X , which is the maximal G -invariant measurable subset of S n on which G actsconservatively, coincides with the big horospheric limit set L H ( G ) up to null setsof µ .We are interested in the difference L H ( G ) \ L h ( G ), which contains all Garnettpoints originally defined in [29]. For the spherical Lebesgue measure, L H ( G ) \ L h ( G )is a null set, but Tukia [36] asked how small this difference is as measured by a G -invariant conformal measure. In this section, we will explain why one shouldexpect that the difference between the big horospheric and the horospheric limitset is small.First we introduce a continuous family of limit sets of a Kleinian group usingthe approaching order of its orbits. Fix c > κ ∈ [0 , z ∈ B n +1 ,let S ( z : c, κ ) be the shadow of a hyperbolic ball B (cid:18) z, κ κ d (0 , z ) + c (cid:19) KURT FALK AND KATSUHIKO MATSUZAKI w.r.t. the projection from the origin to S n (see [12] for more details). Essentiallythe same shadow I ( z : c, α ) := (cid:26) ξ ∈ S n (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ξ − z | z | (cid:12)(cid:12)(cid:12)(cid:12) < c (1 − | z | ) α (cid:27) was used in Nicholls [20]; I ( z : c, α ) corresponds to S ( z : c, κ ) via α = 1 / (1+ κ ). Fora Kleinian group G acting on B n +1 , consider the orbit Gz = { gz } g ∈ G of z ∈ B n +1 and define L ( κ ) r ( G ) := [ c> lim sup g ∈ G S ( gz : c, κ ) . This is the set of points ξ ∈ S n such that ξ belongs to infinitely many S ( gz : c, κ )for some c >
0. It is not difficult to see that L ( κ ) r ( G ) is independent of the choiceof z .When κ = 0 the set L ( κ ) r ( G ) is nothing more than the radial limit set L r ( G ),and when κ = 1, L ( κ ) r ( G ) coincides with the big horospheric limit set L H ( G ). Bymoving κ between 0 and 1, we are thus interpolating between the radial limit setand the horospheric limit set.Related limit sets were introduced by Bishop [4] and Lundh [16]. We set ϕ ξ ( t ) := d ( g ξ,z ( t ) , g ξ,z (0)) , which is the hyperbolic distance in the quotient manifold M G between g ξ,z ( t ) andthe initial point g ξ,z (0). Alternatively, it is defined as the distance of the orbit Gz from ˜ g ξ,z ( t ) in B n +1 . It is clear that ϕ ξ ( t ) ≤ t . The ratio ϕ ξ ( t ) /t measures howrapidly or slowly the geodesic ray g ξ,z ( t ) escapes to infinity as t → ∞ . For instance,in Bishop [4], g ξ,z ( t ) is called a linearly escaping geodesic if there exists a positiveconstant κ > ϕ ξ ( t ) /t > κ for all t . However, here we mainly investigategeodesic rays that are escaping to infinity slowly.For each κ ∈ [0 ,
1] we define the following limit set as the set of end points ofsublinearly escaping geodesic rays:Λ κ ( G ) := { ξ ∈ S n | lim inf t →∞ ϕ ξ ( t ) t ≤ κ } . The radial limit points correspond to non-escaping geodesic rays and hence L r ( G )is contained in the sublinear growth limit set Λ ( G ). As an important extremalcase, we consider the strong sublinear growth limit set Λ ∗ ( G ) := { ξ ∈ S n | lim t →∞ ϕ ξ ( t ) t = 0 } , which is contained in Λ ( G ). However, while clearly L M ( G ) ⊂ L r ( G ) ⊂ Λ ( G ),the inclusion relation of L r ( G ) or L M ( G ) to Λ ∗ ( G ) is not a priori clear (see e.g.Example 6.2).As it turns out, the limit sets L ( κ ) r ( G ) and Λ κ ( G ), while not being coincident,are very similar. Actually, Lundh [16, Th.4.1] proved that L ( κ ) r ( G ) = (cid:26) ξ ∈ S n (cid:12)(cid:12)(cid:12)(cid:12) lim inf t →∞ (cid:18)
11 + κ ( ϕ ξ ( t ) + t ) − t (cid:19) < ∞ (cid:27) for 0 ≤ κ <
1, and moreover, \ c> lim sup g ∈ G I ( gz : c, (1 + κ ) − ) = (cid:26) ξ ∈ S n (cid:12)(cid:12)(cid:12)(cid:12) lim inf t →∞ (cid:18)
11 + κ ( ϕ ξ ( t ) + t ) − t (cid:19) = −∞ (cid:27) OROSPHERIC LIMIT SETS 7 for 0 < κ ≤
1. As a consequence, it was shown in [16, Cor.4.2] that L ( κ ) r ( G ) ⊂ Λ κ ( G )is always valid, and if κ ′ < κ then Λ κ ′ ( G ) ⊂ L ( κ ) r ( G ).When κ = 1, Λ ( G ) coincides with the entire sphere S n since ϕ ξ ( t ) /t ≤ ξ ∈ S n and all t >
0. Since the horospheric limit set L h ( G ) can be represented by T c> lim sup g ∈ G I ( gz : c, / Proposition 3.1. L h ( G ) = { ξ ∈ S n | lim inf t →∞ ( ϕ ξ ( t ) − t ) = −∞} . However, a similar dynamical description of the big horospheric limit set L H ( G )is somewhat more involved and we shall deal with this in Section 4.It is well known [5] that, for any non-elementary Kleinian group G , the Haus-dorff dimension of the radial limit set L r ( G ) = L (0) r ( G ) coincides with the criticalexponent δ ( G ). The elementary estimate of the Hausdorff dimension here is theone from above, and the corresponding argument can be generalised to prove anupper bound for the Hausdorff dimension dim H L ( κ ) r ( G ) as follows. This was provedin [12, p.575]. See also [4, Cor.3] and [6, Prop.4.2] for versions of this statementformulated for the limit sets Λ κ ( G ). Proposition 3.2.
A Kleinian group G satisfies dim H L ( κ ) r ( G ) ≤ (1 + κ ) δ ( G ) for every κ ∈ [0 , . We have already mentioned that L (1) r ( G ) = L H ( G ) and Λ ( G ) = S n . However,when κ <
1, the limit sets L ( κ ) r ( G ) and Λ κ ( G ) are contained in L h ( G ) as thefollowing proposition asserts. Proposition 3.3.
For any Kleinian group G we have [ ≤ κ< L ( κ ) r ( G ) ⊂ L h ( G ) ⊂ L (1) r ( G ) = L H ( G ) . Proof.
By the relationship between L ( κ ) r ( G ) and Λ κ ( G ), we see that S ≤ κ< L ( κ ) r ( G )equals S ≤ κ< Λ κ ( G ). For any point ξ ∈ Λ κ ( G ) with 0 ≤ κ <
1, its definition giveslim inf t →∞ ϕ ξ ( t ) t ≤ κ < . Since ϕ ξ ( t ) is Lipschitz continuous, this implies in particular thatlim t →∞ ( ϕ ξ ( t ) − t ) = −∞ . By Proposition 3.1, we conclude that ξ ∈ L h ( G ). (cid:3) The statement of Proposition 3.3 goes to show that the difference between L h ( G )and L H ( G ) is so small that it cannot be detected by the stratification of the limitset given by the family of sets L ( κ ) r ( G ), κ > KURT FALK AND KATSUHIKO MATSUZAKI Dynamical characterisation of the big horospheric limit set
In the previous section we have seen a dynamical characterisation of horosphericlimit points ξ ∈ L h ( G ) in terms of the distance function ϕ ξ ( t ) along the geodesic raytowards ξ . The corresponding result for a big horospheric limit point ξ ∈ L H ( G )does not have such a neat form, but we can think of the following claim for L H ( G )as having a similar flavour as in the case of L h ( G ).For a geodesic ray ˜ g ξ,z in B n +1 , the Busemann function b ( x ) for x ∈ B n +1 isdefined by b ( x ) := lim t →∞ { d ( x, ˜ g ξ,z ( t )) − t } . Note that the limit always exists since the function taken the limit is bounded frombelow and decreasing. A horosphere tangent at ξ is a level set of b ( x ). For instance,the horosphere passing through z is given by { x ∈ B n +1 | b ( x ) = 0 } . Proposition 4.1.
For a Kleinian group G and fixed z ∈ B n +1 , the point ξ ∈ S n belongs to L H ( G ) if and only if there exists a sequence t < t < t < . . . converging to infinity and a constant M such that for each n ∈ N there is a geodesicsegment β n connecting g ξ,z (0) and g ξ,z ( t n ) of length not greater than t n + M suchthat the closed curves in the family g ξ,z ([0 , t n ]) ∪ β n , n ≥ , are mutually freelynon-homotopic to each other.Proof. Assume that ξ ∈ L H ( G ). Then there is a horosphere H and a sequence ( γ n )in G such that the γ n ( z ) are inside of the horoball bounded by H and converge to ξ as n → ∞ . We write H = { x ∈ B n +1 | b ( x ) = M ′ } for some M ′ by using theBusemann function b for ˜ g ξ,z . Set M = M ′ + ε for some ε >
0. By the definitionof the Busemann function, we can find t n > n such that d ( γ n ( z ) , ˜ g ξ,z ( t n )) − t n ≤ M. By taking the projection of the geodesic connecting γ n ( z ) and ˜ g ξ,z ( t n ) to M G as β n , we see that this family of closed curves satisfies the required condition.Conversely, if we have such a sequence of geodesics β n on M G , we lift them to B n +1 so that, for each n , one end point is ˜ g ξ,z ( t n ) on a fixed geodesic ray ˜ g ξ,z towards ξ . Then the other end point of the lift of β n is an orbit of z by G , whichlies inside of the horosphere tangent at ξ given by b ( x ) = M . Hence we have that ξ ∈ L H ( G ). (cid:3) As an application of this claim, we can easily explain the result in [17, Th.6]corresponding to Theorem 5.1, which asserts the inclusion relation L r ( G ) ⊂ L H ( N )for a non-trivial normal subgroup N of a non-elementary Kleinian group G . Indeed,for ξ ∈ L r ( G ), choose a geodesic ray ˆ g ξ,z in M G that returns infinitely often to somebounded neighbourhood of the initial point. Then its lift g ξ,z to M N travels withina bounded distance along preimages under the covering transformation M N → M G of some fixed closed geodesic. If we make a detour at one of these closed geodesics,we can find a geodesic β n as in Proposition 4.1. We can do this infinitely often inany tail of the geodesic ray, and so ξ ∈ L H ( N ).By a similar argument, we have the following consequence from Proposition 4.1. Proposition 4.2.
Let G be a Kleinian group such that the convex core C ( M G ) of M G = B n +1 /G has bounded geometry, that is, there is a constant M > such thatthe injectivity radius at every point of C ( M G ) is bounded by M . Then every limitpoint ξ ∈ L ( G ) belongs to L H ( G ) . OROSPHERIC LIMIT SETS 9
It was remarked in [11, Prop.5.1] that any G -invariant conformal measure µ doesnot have an atom at ξ ∈ L H ( G ) unless ξ is a parabolic fixed point. In particular,from the above proposition, we see that a non-parabolic ending measure for G hasno atom if the convex core C ( M G ) has bounded geometry.5. The Myrberg limit set is contained in the horospheric limit set ofa normal subgroup
In this section we answer Tukia’s question formulated in the introduction and thebeginning of Section 3 about measuring the difference between the big horosphericand the horospheric limit sets for normal subgroups of Kleinian groups of divergencetype, and also give a conjecture generalising the statement in this case. The firsttheorem is the main step towards the answer and is interesting in itself. Beforestating it, we need a few preparations.A point ξ ∈ L ( G ) is a Myrberg limit point of G if for any distinct limit points x, y ∈ L ( G ) and for any geodesic ray β towards ξ there is a sequence { g n } ⊂ G suchthat g n ( β ) converges to the geodesic line connecting x and y . The set of all Myrberglimit points of G is called the Myrberg limit set and is denoted by L M ( G ). Theidea originated in [18] and was introduced as a qualitative version of ergodicity ascharacterised by Birkhoff’s ergodic theorem. Further developments can be foundin [34], [19] and [2], and state-of-the-art papers are [35], [27] and [8]. The mainresult in the latter three papers is that a Kleinian group being of divergence type isequivalent to its Myrberg limit set having full Patterson measure. All geometricallyfinite Kleinian groups thus have their Myrberg limit set of full measure.We can also define the Myrberg limit set by using the geodesic flow on T M G .Denote the closed subset of unit tangent vectors that generate geodesic lines stayingin the convex core by V C = { ( v, p ) ∈ T M G | g v,p ( t ) ∈ C ( M G ) for all t ∈ R } . Then ξ ∈ L M ( G ) if and only if, for p ∈ C ( M G ) and v ξ being the projection ofthe tangent vector pointing towards ξ based at some lift of p , the forward orbit { φ t ( v ξ , p ) | t ∈ R } of ( v ξ , p ) under the geodesic flow contains unit tangent vectorsthat are arbitrarily close to any element of V C .We consider normal subgroups N of a Kleinian group G and how properties oflimit sets are inherited from G to N . In [17, Th.6], we have seen the inclusionrelation L r ( G ) ⊂ L H ( N ). In the present paper, as a refinement of this argument,we prove the following theorem. Theorem 5.1.
Let N be a non-trivial normal subgroup of the Kleinian group G .Then, L M ( G ) ⊂ L h ( N ) . We first give a geometric explanation in the manifolds. Since N is a non-trivialnormal subgroup of G , it is non-elementary and hence it contains a loxodromicelement h . Let c be the closed geodesic in M N corresponding to h and ˆ c theprojection of c under the normal covering M N → M G , which may be a multi-curve.Take any Myrberg limit point ξ ∈ L M ( G ) and a geodesic ray in B n +1 starting from z ∈ B n +1 and towards ξ ∈ S n . Then its projection to M G follows ˆ c infinitely manytimes within an arbitrarily small tubular neighbourhood.We consider a lift of this geodesic ray to M N , which is denoted by g ξ,z ( t ) ( t ≥ c under the covering transformations of ξ oxx’ kk kk k−1 k g n g ([x’ , x ])g ([x’ , x ]) k kkk Figure 1.
The setting of Theorem 5.1. M N → M G infinitely many times within arbitrarily small tubular neighbourhoods.This implies that once g ξ,z ( t ) turns around a copy of c , we can find a geodesicbetween g ξ,z (0) and g ξ,z ( t ) which is shorter than t by some uniform length ℓ > ϕ ξ ( t ) = d ( g ξ,z ( t ) , g ξ,z (0)) satisfies ϕ ξ ( t ) ≤ t − ℓ at the time t when wefinish one round. However, since such detours occur infinitely many times, we havethat lim t →∞ ( ϕ ξ ( t ) − t ) = −∞ . By Proposition 3.1, this shows that ξ ∈ L h ( N ). Proof of Theorem 5.1.
Consider some arbitrary ξ ∈ L M ( G ). G is assumed to benon-elementary and N to be non-trivial, so N will always contain hyperbolic ele-ments. Let n ∈ N be one of these and consider the uniquely determined point o on its axis so that the geodesic ray [ o, ξ ) from o to ξ is orthogonal on the axis of n .For all k ∈ N put x k := n k ( o ) and x ′ k := n − k ( o ) and denote the geodesic segmentconnecting x ′ k and x k by [ x ′ k , x k ].Given some arbitrary but fixed ε >
0, we have by the Myrberg property of ξ that there is a sequence ( g k ) k ∈ N of elements of G so that g k ( o ) tends to ξ in theEuclidean metric and, for all k ∈ N , the geodesic segment g k ([ x ′ k , x k ]) is ε -close to[ o, ξ ), meaning that any point on g k ([ x ′ k , x k ]) is within distance ε from the geodesicray [ o, ξ ).Since N is normal in G , we know that g k n k g − k , g k n − k g − k ∈ N for all k ∈ N .A priori, it is not clear how g k ([ x ′ k , x k ]) is ‘oriented’ with respect to [ o, ξ ), but weshall see that this is not important for the argument. For a given k ∈ N , assuming g k ( x ′ k ) is closer to o than g k ( x k ), we have that d ( g k n k g − k ( x k ) , g k ( o )) = d ( n k g − k ( x k ) , o ) = d ( n k g − k ( x k ) , n k ( x ′ k ))= d ( g − k ( x k ) , x ′ k ) = d ( x k , g k ( x ′ k )) ≍ + d ( o, g k ( o )) OROSPHERIC LIMIT SETS 11 and that d ( g k n k g − k ( x ′ k ) , g k ( x k )) = d ( n k g − k ( x ′ k ) , x k ) = d ( n k g − k ( x ′ k ) , n k ( o ))= d ( g − k ( x ′ k ) , o ) = d ( x ′ k , g k ( o )) ≍ + d ( o, g k ( x k )) . Here, the additive comparabilities ≍ + , which mean that the difference between thecomparable distances is uniformly bounded independently of k , are due to the factthat [ o, ξ ) is orthogonal on the axis of n and g k ([ x ′ k , x k ]) is ε -close to [ o, ξ ). If g k ( x k )is closer to o than g k ( x ′ k ), then the same argument as above yields that d ( g k n − k g − k ( x ′ k ) , g k ( o )) ≍ + d ( o, g k ( o )); d ( g k n − k g − k ( x k ) , g k ( x ′ k )) ≍ + d ( o, g k ( x ′ k )) . From these estimates, elementary hyperbolic geometry shows that there is somehorosphere H tangent at ξ such that either both g k n k g − k ( x k ) and g k n k g − k ( x ′ k ) orboth g k n − k g − k ( x k ) and g k n − k g − k ( x ′ k ) lie inside of H for all k . Since d ( g k n k g − k ( x k ) , g k n k g − k ( x ′ k )) = d ( g k n − k g − k ( x k ) , g k n − k g − k ( x ′ k )) = d ( x k , x ′ k ) → ∞ as k → ∞ , the mid points g k n k g − k ( o ) or g k n − k g − k ( o ) of the geodesic segmentsenter smaller and smaller horospheres tangent at ξ . This can be easily seen if weuse the upper half-space model of the hyperbolic space with ξ being infinity. Hencewe have that ξ ∈ L h ( N ). (cid:3) As a direct corollary to Theorem 5.1, we obtain that the difference between thebig horospheric and the horospheric limit sets of N is a null set for the Pattersonmeasure µ for G when G is of divergence type. Also, since N is normal in G , wehave that L ( N ) = L ( G ) and that µ is a conformal measure of dimension δ ( G ) for N as well. Note that δ ( N ) may very well be strictly smaller than δ ( G ), in particular,when G is convex cocompact and G/N is non-amenable ([7]). Concerning theinvestigation of this phenomenon in view of the dimension gap between L ( N ) and L r ( N ), see for instance [10] or the survey [9]. Theorem 5.2.
Let N be a non-trivial normal subgroup of the Kleinian group G ,and assume that G is of divergence type. Then, µ δ ( G ) ( L H ( N ) \ L h ( N )) = 0 for some N -invariant conformal measure µ δ ( G ) of dimension δ ( G ) . The following class of examples illustrates the statements of Theorem 5.1 andTheorem 5.2 in a non-trivial way.
Example 5.3.
Let G and G be Schottky groups with fundamental domainshaving disjoint complements in hyperbolic space, and define G := G ∗ G which isthen also a Schottky group. Put N := ker( ϕ ), where ϕ : G → G is the canonicalgroup homomorphism. Thus, 0 → N → G → G → N is the normal subgroup of G generated by G in G , and G/N ∼ = G . Clearly, N is infinitely generated and if we assume that G is freely generated by at leasttwo generators, and is thus non-amenable, then the already mentioned result ofBrooks [7] ensures that δ ( N ) < δ ( G ). For more details on this class of examples seealso [12]. Theorem 5.1 now applies and we thus have that L M ( G ) ⊂ L h ( N ). Wealso know on the one hand that in this situation the Hausdorff dimension of L M ( G )coincides with δ ( G ) since G is cocompact and thus L M ( G ) is of full measure w.r.t.the Patterson measure of G ([35], [27] and [8]) which is known [28] to be proportional to the δ ( G )-dimensional Hausdorff measure on L ( G ). On the other hand, we knowby [5] that dim H ( L r ( N )) = δ ( N ). We thus know that L r ( N ) has strictly smallerHausdorff dimension than both L M ( G ) and L h ( N ) which makes the statements ofboth Theorem 5.1 and Theorem 5.2 meaningful and non-trivial.Concerning the difference between the big and the small horospheric limit setsin Theorem 5.2, we are considering the situation where N is contained as a normalsubgroup in some Kleinian group G , and the limit sets are measured by a conformalmeasure for G . However, it is desirable to describe the difference between these limitsets only by using the Kleinian group in question itself. Here is an idea how to dothis, which makes use of our previous work [10].We call a discrete G -invariant set X = { x i } ∞ i =1 in the convex hull H ( L ( G )) of L ( G ) uniformly distributed if the following two conditions are satisfied:(i) There exists a constant M < ∞ such that, for every point z ∈ H ( L ( G )),there is some x i ∈ X such that d ( x i , z ) ≤ M ;(ii) There exists a constant m > x i and x j in X satisfy d ( x i , x j ) ≥ m .For a uniformly distributed set X , we define the extended Poincar´e series withexponent s > z ∈ B n +1 by P s ( X, z ) := X x ∈ X e − s d ( x,z ) . The critical exponent for X is∆ := inf { s > | P s ( X, z ) < ∞} . The Poincar´e series for X is of convergence type if P ∆ ( X, z ) < ∞ , and of divergencetype otherwise. Moreover, we can define the associated Patterson measure µ X for X supported on L ( G ) by a similar construction to the usual case.As a sufficient condition for the extended Poincar´e series P s ( X, z ) to be of di-vergence type, we have the following. A uniformly distributed set X is of boundedtype if there exists a constant ρ ≥ X ∩ B R ( x )) X ∩ B R ( z )) ≤ ρ for every x ∈ X and for every R >
0. In this case, the ∆-dimensional Hausdorffmeasure of L ( G ) is positive and P ∆ ( X, z ) = ∞ . For more details see [10].In view of these similarities to the case where our group in question is a nor-mal subgroup of some Kleinian group of divergence type, we give the followingconjecture in analogy to Theorem 5.2. Conjecture 1. If G is a Kleinian group whose convex hull H ( L ( G )) admits auniformly distributed set X so that the extended Poincar´e series P s ( X, z ) is ofdivergence type, then µ X ( L H ( G ) \ L h ( G )) = 0 for the associated Patterson measure µ X . If L ( G ) = S n , then we can choose the Lebesgue measure on S n as µ X . In thiscase, the original result of Sullivan [29] on Garnett points supports the conjecture. OROSPHERIC LIMIT SETS 13 The Hausdorff dimension of the Myrberg limit set
In this section we justify why in the previous section we considered conformalmeasures of dimension δ ( G ) in order to measure the difference between the bighorospheric and horospheric limit sets of N . Namely, we will show under a certainassumption that the Hausdorff dimension of L M ( G ), and thus, in view of Theo-rem 5.1, of both L h ( N ) and L H ( N ), is equal to δ ( G ). Proposition 6.1. If G is a Kleinian group of divergence type such that the strongsublinear growth limit set Λ ∗ ( G ) has full measure for the Patterson measure µ of G , then dim H ( L M ( G )) = δ ( G ) . All assumptions on G follow from the condition µ ∗ ( T M G ) < ∞ . The method of proof is a generalisation of the argument for the radial limit setgiven in Sullivan [28]. (See also Nicholls [20, Th.9.3.5].) Note that Sullivan alreadyconjectures in the original paper that for any divergence type group G the strongsublinear growth limit set Λ ∗ ( G ) is of full measure for the Patterson measure of G .That is, Proposition 6.1 should be valid without the extra assumption on Λ ∗ ( G ).One can justifiably ask why this is not clear for divergence type groups in general.The reason is that for geometrically infinite groups G , the radial limit set L r ( G )is not necessarily contained in Λ ∗ ( G ) as the following example shows. However,we expect that L M ( G ) ⊂ Λ ∗ ( G ) should be true. This can still be regarded as ageneralisation of Sullivan’s conjecture. Example 6.2.
Let T be a once-punctured torus, and let a and b be simple closedgeodesics on T whose intersection number is 1. We cut open T along a to obtain abordered surface P with one puncture and two geodesic boundary components.We prepare infinitely many copies of P and paste them one after another alongthe geodesic boundary components without a twist. The resulting surface is a cycliccover of T , which is denoted by R . Let h h i be the covering transformation group.The lift of b to R , which is a geodesic line invariant under h h i , is denoted by ˜ b . Wealso take a simple closed geodesic a that is a component of the lift of a and setthe intersection of ˜ b and a as a base point o . Set a n = h n ( a ) and o n = h n ( o ) forevery integer n .We consider the following infinite curve starting at o : β = ∞ Y k =0 (˜ b [ o, o ( − k ] · a ( − k · ˜ b [ o ( − k , o ]) . Here, ˜ b [ x, y ] denotes the segment in ˜ b from x to y . Then, we take the geodesic ray β : [0 , ∞ ) → R starting from o and going to infinity navigated homotopically by β .Since β returns infinitely many times to some neighbourhood of o , the end point of β gives a radial limit point ξ ∈ L r ( G ) of a Fuchsian group G uniformising R .On the other hand, ξ does not belong to the strong sublinear growth limit setΛ ∗ ( G ) of G . To see this, for every n ≥
1, let t n > β = g ξ,o such that β ( t ) crosses over a ( − n − ( − n for the firsttime. We denote the hyperbolic length by ℓ ( · ) and the hyperbolic distance by d ( · , · ). o o o o o oo b β aaaaaaa
00 1 2 3 44321−1−2−2 −1 ~
Figure 2.
The setting of Example 6.2.Then, we have that t n ≤ n − X k =0 { ℓ ( e b [ o, o ( − k ]) + ℓ ( a ) } + ℓ ( e b [ o, o ( − n ]) + ℓ ( a ) ≤ · n ℓ ( b ) + ( n + 1) ℓ ( a ) < · n { ℓ ( b ) + ℓ ( a ) } . However, since d ( o, β ( t n )) ≥ (2 n − d ( a , a ), we obtain that d ( o, β ( t n )) t n > d ( a , a ) ℓ ( b ) + ℓ ( a )for every n ≥
1. This implies thatlim sup t →∞ ϕ ξ ( t ) t > , and hence ξ / ∈ Λ ∗ ( G ). Proof of Proposition 6.1.
It is proved in [35], [27] and [8] that for a Kleinian group G of divergence type, the Myrberg limit set L M ( G ) has full measure w.r.t. thePatterson measure µ of G . Moreover, by assumption, the sublinear growth limitset Λ ∗ ( G ) has full µ -measure.Following the argument from [20, Lemma 9.3.4], we can find a compact subset K of L M ( G ) ∩ Λ ∗ ( G ) with µ ( K ) > ε > r > ξ ∈ K and r < r then µ ( B ( ξ, r ) ∩ K ) /r δ ( G ) − ε < A, where A is some absolute constant. From this property, it follows that K haspositive ( δ ( G ) − ε )-dimensional Hausdorff measure for any ε > H L M ( G ) ≥ δ ( G ). The converse inequality is clear from dim H L r ( G ) = δ ( G ) and L M ( G ) ⊂ L r ( G ), and thus the first statement follows.To verify the latter statement, it suffices to remark that µ ∗ ( T M G ) < ∞ impliesthat G is of divergence type and that Λ ∗ ( G ) has full µ -measure (see [28, Cor.19],[20, Lemma 9.3.3]). (cid:3) Example 6.3.
Here it is interesting to mention a class of non-trivial examples forwhich the statement of Proposition 6.1 applies. In [23] Peign´e constructs geometri-cally infinite Schottky groups G of divergence type which at the same time satisfy OROSPHERIC LIMIT SETS 15 µ ∗ ( T M G ) < ∞ . For more details we refer the interested reader to the originalarticle [23].Following a completely different idea of proof, it may be possible to generalisethe argument in Bishop and Jones [5] showing that the the Hausdorff dimensionof the radial limit set coincides with the critical exponent, in order to prove thefollowing conjecture. Conjecture 2.
For any non-elementary Kleinian group G we have dim H ( L M ( G )) = δ ( G ) . References [1] J. Aaronson and D. Sullivan, ‘Rational ergodicity of geodesic flows’,
Ergod. Th. & Dynam.Sys. (1984), no. 2, 165–178.[2] S. Agard, ‘A geometric proof of Mostow’s rigidity theorem for groups of divergence type’, Acta Math. (1983), no. 3-4, 231–252.[3] J. W. Anderson, K. Falk and P. Tukia, ‘Conformal measures associated to ends of hyperbolic n -manifolds’, Quart. J. Math. (2007), 1–15.[4] C. J. Bishop, ‘The linear escape limit set’, Proc. Amer. Math. Soc. (2003), 1385–1388.[5] C. J. Bishop and P. Jones, ‘Hausdorff dimension and Kleinian groups’,
Acta Math. (1997),1–39.[6] P. Bonfert-Taylor, K. Matsuzaki and E. C. Taylor, ‘Large and small covers of a hyperbolicmanifold’,
J. Geom. Anal. (2012), 455–470.[7] R. Brooks, ‘The bottom of the spectrum of a Riemannian covering’, J. Reine Angew. Math. (1985), 101–114.[8] K. Falk, ‘A note on Myrberg points and ergodicity’,
Math. Scand. (2005), 107–116.[9] K. Falk, ‘Dimensions of limit sets of Kleinian groups’, to appear in the conference proceedings Horizons in Fractal Geometry and Complex Dimensions , book series of Contemp. Math.[10] K. Falk and K. Matsuzaki, ‘The critical exponent, the Hausdorff dimension of the limit setand the convex core entropy of a Kleinian group’,
Conform. Geom. Dyn. (2015), 159–196.[11] K. Falk, K. Matsuzaki and B. O. Stratmann, ‘Checking atomicity of conformal ending mea-sures for Kleinian groups’, Conform. Geom. Dyn. (2010), 167–183.[12] K. Falk and B. O. Stratmann, ‘Remarks on Hausdorff dimensions for transient limit sets ofKleinian groups’, Tohoku Math. J. (2) (2004), 571–582.[13] E. Hopf, ‘Fuchsian groups and ergodic theory’, Trans. Amer. Math. Soc. (1936), 299–314.[14] E. Hopf, ‘Ergodic theory and the geodesic flow on surfaces of constant negative curvature’, Bull. Amer. Math. Soc. (1971), no. 6, 863–877.[15] V. A. Kaimanovich, ‘Hopf decomposition and horospheric limit sets’, Ann. Acad. Sci. Fenn. (2010), 335–350.[16] T. Lundh, ‘Geodesics on quotient manifolds and their corresponding limit points’, MichiganMath. J. (2003), 279–304.[17] K. Matsuzaki, ‘Conservative action of Kleinian groups with respect to the Patterson-Sullivanmeasure’, Comp. Meth. Funct. Th. (2002), 469–479.[18] P.J. Myrberg, ‘Ein Approximationssatz fur die Fuchsschen Gruppen’, Acta Math. (1931),389–409.[19] T. Nakanishi, ‘On P.J. Myrberg’s approximation theorem for some Kleinian groups’, Journ.Math. Kyoto. Univ. (1985), no. 3, 405–419.[20] P. J. Nicholls, The ergodic theory of discrete groups , LMS Lecture Notes Series 143, Cam-bridge Univ. Press, 1989.[21] S. J. Patterson, ‘The limit set of a Fuchsian group’,
Acta Math. (1976), 241–273.[22] S. J. Patterson, ‘Lectures on limit sets of Kleinian groups’, in
Analytical and geometric aspectsof hyperbolic space , 281–323, Cambridge University Press, 1987.[23] M. Peign´e, ‘On the Patterson-Sullivan measure of some discrete group of isometries’,
IsraelJ. Math. (2003), 77–88. [24] M. Rees, ‘Checking ergodicity of some geodesic flows with infinite Gibbs measure’,
Ergod.Th. & Dynam. Sys. (1981), 107–133.[25] T. Roblin, ‘Sur l’ergodicit´e rationelle et les propri´et´es ergodiques du flot g´eod´esique dans lesvari´et´es hyperboliques’, Ergod. Th. & Dynam. Sys. (2000), 1785–1819.[26] T. Roblin, ‘Sur la fonction orbitale des groupes discrets en courbure n´egative’, Ann. Inst.Fourier (Grenoble) (2002), 145–151.[27] B. O. Stratmann, ‘A remark on Myrberg initial data for Kleinian groups’, Geometriae Dedi-cata (1997), 257–266.[28] D. Sullivan, ‘The density at infinity of a discrete group of hyperbolic motions’, Inst. Hautes´Etudes Sci. Publ. Math. (1979), 171–202.[29] D. Sullivan, ‘On the ergodic theory at infinity of an arbitrary discrete group of hyperbolicmotions’, Riemann surfaces and related topics , Ann. Math. Studies , Princeton UniversityPress, 465–496, 1981.[30] D. Sullivan, ‘Travaux de Thurston sur les groupes quasi-Fuchsiens et les varietes hyperboliquesde dimension 3 fibrees sur S ’, S´eminaire Bourbaki, 32e ann´ee, Vol. 1979/80, Exp. 554, Lect.Notes Math. no. 842, Springer Verlag, 1981, 196–214.[31] D. Sullivan, ‘Entropy, Hausdorff measures old and new, and limit sets of geometrically finiteKleinian groups’, Acta Math. (1984), 259–277.[32] D. Sullivan, ‘Related aspects of positivity in Riemannian geometry’,
J. Diff. Geom. (1987),327–351.[33] W.P. Thurston, The Geometry and Topology of Three-Manifolds , lecture notes, PrincetonUniv., Princeton, NJ, 1979.[34] M. Tsuji, ‘Myrberg’s approximation theorem for Fuchsian groups’,
Journ. Math. Soc. Japan (1952), 310–312.[35] P. Tukia, ‘The Poincar´e series and the conformal measure of conical and Myrberg limit points’, J. Anal. Math. (1994), 241–259.[36] P. Tukia, ‘Conservative action and the horospheric limit set’, Ann. Acad. Sci. Fenn. (1997), 387–394. Christian-Albrechts-Universit¨at zu Kiel, Mathematisches Seminar, Ludewig-Meyn-Str. 4, 24118 Kiel, Germany
E-mail address : [email protected] Department of Mathematics, School of Education, Waseda University, Nishi-Waseda1-6-1, Shinjuku, Tokyo 169-8050, Japan
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