Infinite index subgroups and finiteness properties of intersections of geometrically finite groups

INFINITE INDEX SUBGROUPS AND FINITENESS PROPERTIES OF INTERSECTIONS OF GEOMETRICALLY FINITE GROUPS Boris Apanasov† Abstract. We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup. 1. Introduction In 1954 Howson [H] proved that in a free group, the intersection of two finitely generated subgroups is finitely generated (such groups become known as Howson groups). There are many generalizations of this theorem in both geometric group theory due to A.Karras, D.Solitar, D.I.Moldavanskii, R.Gitik, E.Rips, H.Short, a.o., and in the theory of discrete groups of hyperbolic isometries due to L.Greenberg, B.Maskit, J.Hempel, J.Anderson, P.Susskind, G.Swarup, T.Soma, W.Thurston, a.o. Many of these results have geometric or topological content, see [An] for a useful survey and bibliography. One of the initial objects, surface groups correspond to discrete groups of isometries of the hyperbolic plane H2. They are finitely generated if and only if they are geometrically finite. Because of that most of results about Howson groups of hyperbolic isometries are for geometrically finite groups (see Section 2 for precise definition), which have very nice topological and geometrical properties. For example, they are finitely presented, and in fact are of type Fmfor m≥1. It was natural to wonder whether two geometrically finite subgroups of a discrete group of hyperbolic isometries have a geometrically finite intersection. Susskind [Su1] showed that Kleinian groups on 2-plane have this property, and later Susskind and Swarup [SS] †Supported in part by the University of Oklahoma Faculty Senate Award and Centre de Recerca Matematica at Barcelona 1 2 BORIS APANASOV showed that any discrete group of isometries of the real hyperbolic space Hn also has it. However, as it was recently shown by Susskind [Su2], without the ambient discreteness condition the intersection of two geometrically finite subgroups of Isom H4 Rmay be an infinitely generated Fuchsian subgroup of Isom H2 R⊂Isom H4 Rwith a circle as the limit set. In this paper, we explore the possible finiteness properties of the intersection of two geometrically finite groups of isometries in negatively curved rank one symmetric spaces including the real hyperbolic ones. Our goal is to provide a simple method for producing geometrically finite groups in various negatively curved rank one symmetric spaces whose intersection is as big and wild as one may need. In particular, such intersection subgroup may have one or more of the following properties: (1) its limit set coincides with the limit sets of initial geometrically finite groups and is arbitrary big or wild, or (2) it is not finitely generated, or (3) it is finitely generated but not geometrically finite, or (4) it is finitely generated but not finitely presented. Briefly, our method of generating examples is to start with a geometrically finite subgroup Gof the isometry group of some rank one symmetric space Xwhich preserves a totally geodesic subspace XG⊂Xand has the following additional structure. There is a short exact sequence 1→G∗→G→A→1 (1.1) where G∗is a group with one of the properties mentioned above, and Ais an infinite Abelian group, in particular, cyclic group Z. Then one applies a twist construction to obtain another geometrically finite group GT⊂ Isom Xwhose intersection with Gis isomorphic to G∗. We feel that there should be examples with intersection which is finitely presented but not of type F3. The problem of finding such groups can be reduced to a realization problem of corresponding groups constructed in combinatorial group theory. In fact beginning with the work of Stallings [S] and Bieri [Bi], there have been many constructions of groups Gwhich fit into the short exact sequence above with A=Zand with a kernel G∗of type Fnbut not Fn+1 for all n≥2. Among such groups there are Gromov hyperbolic groups [G1, G2], CAT(0) and cubical groups Gwith kernel G∗in (1.1) of type F2(finitely presented) but not of type F3. It is an interesting challenge for the future to see if any of these groups can be realized as subgroups of a geometrically finite group of isometries of some rank one symmetric space. This work was inspired by a talk of Perry Susskind in Spring 2000 at Department of Mathematics of the University of Oklahoma, whose results INTERSECTIONS OF GEOMETRICALLY FINITE GROUPS 3 were later published in [Su2]. Author gratefully acknowledges a fruitful discussion with Perry Susskind and Noel Brady following that talk, as well as inspiring discussions in Centre de Recerca Matematica at Barcelona, especially with Oleg Bogopolski and Martin Lustig. 2. Background and preliminary results Here we fix some notations and definitions, and briefly review some facts concerning geometry of negatively curved symmetric spaces Xof rank one, Carnot groups Nwhich correspond to horospheres in these spaces, see [AX2], as well as geometrical finiteness in such spaces [A1, A2, B], including some constructions of discrete geometrically infinite groups in hyperbolic spaces. 2.1. Symmetric spaces of rank one. There are models of negatively curved symmetric spaces X(or equivalently, F-hyperbolic spaces Hn F) in the unit balls Bn F(0,1) ⊂Fnwhere Fdenotes either real and complex numbers, Rand C, or quaternions H, or Cayley numbers O(octonions, in which case n= 2). The group of orientation preserving isometries of the real hyperbolic space Hn Rwith sectional curvature K≡ −1 is given by PSO(n, 1), and the holomorphic automorphism groups of other spaces with variable sectional curvature K, −1≤K≤ −1/4 respectively are PU(n, 1), PSp(n, 1), and F−20 4. The second components of the isometry groups Isom Xof these complex spaces Xhave anty-holomorphic isometries (compositions of the holomorphic ones and the operator of conjugation in Fn). Isometries g∈Isom Xin these spaces may have one of the following three types which exhaust all the possibilities. If gfixes a point in X, it is called elliptic. If ghas exactly one fixed point, and it lies in ∂X,gis called parabolic. If ghas exactly two fixed points, and they lie in ∂X,gis called loxodromic. A subgroup G⊂Isom Xis called discrete if it is a discrete subset of Isom X. An infinite discrete group Gis called parabolic if it has exactly one fixed point {p}= fix(G), p∈∂X; then Gconsists of either parabolic or elliptic elements. We denote by Λ(G) the limit set of G, i.e. the set of all accumulation points of the orbit G(x), x∈X, and by Ω(G) the complement ∂X\Λ(G) (the discontinuity set at infinity). Various horospheres Xt⊂X,t∈R, centered at a point ∞ ∈ ∂X can be identified with the nilpotent group Nin the Iwasawa decomposition of Isom X=KAN. In its turn, for Hn F, the nilpotent group Ncan be identified with the product Fn−1×Im Fequipped with the operations: (ξ, v)·(ξ0, v0) = (ξ+ξ0, v+v0+2 Imhξ, ξ0i) and (ξ, v)−1= (−ξ, −v),(2.1) 4 BORIS APANASOV where h,iis the standard Hermitian product in Fn−1,hz, wi=Pziwi. The group Nis a 2-step nilpotent Carnot group with center {0}×Im F⊂Fn−1× Im F, and acts on itself by the left translations Th(g) = h·g , h, g ∈ N. We may identify X\{p}=X∪∂X\{p} → N ×[0,∞), and call this identification the ”upper half-space model” for our symmetric space X. This and the standard coordinates on the nilpotent Carnot group Ngive us horospherical coordinates on X, see [AX2]. In horospherical coordinates on X=Hn F, the left action (2.1) of the Carnot group Non itself extends to an isometric action (Carnot translation) on X=Hn F=Fn−1×Im F×(0,∞) in the following form: T(ξ0,v0): (ξ, v, u)7−→ (ξ0+ξ , v0+v+ 2 Imhξ0, ξi, u),(2.2) where (ξ, v, u)∈Fn−1×Im F×[0,∞). Note that the group K0Afixing the origin of Nand ∞(or equivalently, fixing the geodesic in X=Hn Fending at those two points at ∂X) is U(n− 1) ×R×U(1) in Hn C, Sp(n−1) ×R×Sp(1) in Hn H, and Spin(7) ×Rin H2 O. The isometry group of N=Fn−1×Im Fis the semidirect product NoC where Cis the compact automorphism group of N(Carnot rotations), i.e. U(n−1) if F=C, Sp(n−1) if F=H, and Spin(7) if F=O. 2.2. Geometrical finiteness in rank one symmetric spaces. Now we consider a discrete group G⊂Isom Xacting by isometries in a symmetric rank one space X, or even in more general Hadamard space with negative pinched sectional curvature. Taking its discontinuity set Ω(G)⊂ ∂X, we have the quotient M(G) = (X∪Ω(G))/G, which plays important role in several equivalent definitions of geometrical finiteness, see [A1, A2, B]. First definition (GF1, originally due to A.Beardon and B.Maskit) is in terms of the limit set Λ(G). A discrete group G⊂Isom Xis geometrically finite if its limit set Λ(G) entirely consists of conical limit points and parabolic cusps. Here a limit point z∈Λ(G) is called a conical limit point of G if, for some (and hence every) geodesic ray `⊂Xending at z, there is a compact set K⊂Xsuch that g(`)∩K6=∅for infinitely many elements g∈G. A parabolic fixed point p∈Λ(G) is called a cusp point of Gif the quotient (Λ(G)\ {p})/Gpby the action of its stabilizer Gp={g∈G:g(p) = p}is compact. Another definition of geometrical finiteness (GF2, originally due to A.Marden) is that the quotient M(G) has only finitely many topological ends and each of them can be identified with the end of M(Γ), where Γ is a maximal parabolic subgroup of G. INTERSECTIONS OF GEOMETRICALLY FINITE GROUPS 5 Two other definitions are originally due to W.Thurston. They use the minimal convex retract C(G)⊂X/G, where C(G) is the G-quotient of the minimal convex set in Xcontaining at infinity the limit set Λ(G) (the convex hull of Λ(G)): (GF3): The thick part of the minimal convex retract (=convex core) C(G) of X/G is compact. (GF4): For some ² > 0, the uniform ²-neighborhood of C(G)⊂X/G has a finite volume, and there is a universal bound on the orders of finite subgroups in G. Theorem 2.1. [B] Let Xbe a pinched Hadamard manifold. Then the four definitions GF1, GF2, GF3 and GF4 of geometrical finiteness for a discrete group G⊂Isom Xare all equivalent. In the case of homogeneous spaces X, the situation can be greatly clarified by using the following Apanasov-Xie [AX2] geometric definition of parabolic cusp points and the standard cusp ends of M(G). It is based on the existence of “invariant hocycles” for a parabolic subgroup Gp⊂Gpreserving horospheres Xp⊂Xcentered at the fixed point p∈∂X,Gp={g∈ G:g(p) = p. Such horospheres Xpare represented by a nilpotent Carnot group Nin Iwasawa decomposition Isom X=KAN. Assuming that it has a compact group of automorphisms C, we have that Gp⊂ N oCacts by isometries in N∼ =∂X\{p}and, due to Apanasov-Xie [AX2] theorem, there exists a minimal affine subspace Np⊂ N represented (up to a change of the origin) by a connected Lie subgroup of Nand preserved by the parabolic stabilizer Gpacting there co-compactly. Now for a given r > 0 and some left-invariant metric on N(e.g. Cygan metric ρc), we consider Gp-invariant sets Up,r ={x∈∂X\{p}:ρc(x, Np)≥ 1/r}and call them standard cusp neighborhoods at pof radius r > 0 if they are precisely invariant with respect to Gp⊂G, i.e. γ(Up,r) = Up,r for γ∈Gpand g(Up,r)∩Up,r =∅for g∈G\Gp. Then, due to [AX2], a parabolic fixed point p∈∂X is a cusp point if and only if it has a standard cusp neighborhood Up,r. Clearly, the quotients ˆ Up,r/Gpof such standard cusp neighborhoods provide a basis of the unique end of the quotient M(Gp). Such an end is called a standard parabolic end of M(G). The family Ep= {ˆ Up,r/Gp}of closed suborbifolds in the orbifold M(G) naturally defines the cusp end of M(G) identified by G-orbit of the parabolic cusp point p. It is isometric to a standard cusp end, actually to the end of the almost nilpotent orbifold M(Gp). Due to Apanasov [A4], such almost nilpotent locally symmetric spaces M(Gp) are vector bundles over the compact infranil base Np/Gpand have finite coverings by the product of Euclidean space Rkand a closed nil-manifold Np/G0 p(which is either a torus or the total 6 BORIS APANASOV space of a torus bundle over a torus). This shows the topological finiteness of cusp ends of non-compact geometrically finite orbifolds M(G) (see [A4]). Therefore geometrically finite groups G⊂Isom Xhave finiteness type Fn for all n≥1 which generalizes the notions of finite generation (F1) and finite presentation (F2): Theorem 2.2. Let Xbe a non-compact symmetric space of rank one. Then any geometrically finite discrete group G⊂Isom Xhas type Fnfor all n≥1 and in particular, is finitely presented. 2.3. Some special hyperbolic groups. Here we review some remarkable constructions of discrete hyperbolic groups. 2.3.1. Jørgensen groups. First doubly generated geometrically infinite discrete groups of isometries of H3 Rare due to a remarkable construction by Troels Jørgensen [J] of a closed hyperbolic 3-manifold fibered over the circle. Here we shortly describe the construction, see [KAG]. Consider for m≥2 the numbers λ, ϕ, ρ, x and ygiven by the equalities: λ=eπi 2m,2ϕ= 1+(17 −8 cos(π/m))1 2,2ρ= (ϕ+ 2)1 2+(ϕ−2)1 2, 2x(ϕ−2)1 2= (3 −ϕ)1 2+(−ϕ−1)1 2,2y(ϕ−2)1 2=−(3 −ϕ)1 2+(−ϕ−1)1 2 Let G⊂SL(2,C) be a discrete group acting by isometries in the halfspace model of H3 Rand generated by the matrices: T=µρ0 0ρ−1¶, A =µ−λx −(1 + x2) 1λ−1x¶. Its commutator subgroup G∗⊂Gis generated by Aand B=TA−1T−1A. Consider an Abelian subgroup Γ ⊂Gwhich acts in the half space R3 +by Euclidean similarities, has an invariant axis `={x∈R3 +:x1=x2= 0}, and is generated by K=ABA−1B−1, which is a rotation by angle 2π/m, and by the dilation T. It creates the following set: I={γAγ−1, γBγ−1, γA−1γ−1, γB−1γ−1:γ∈Γ}.(2.1) Now we define a convex hyperbolic polyhedron Q⊂H3 Ras the intersection of hyperbolic half spaces bounded by hyperbolic planes whose boundaries at infinity C∪ {∞} =∂H3 Rare represented by isometric circles {z∈C: INTERSECTIONS OF GEOMETRICALLY FINITE GROUPS 7 |g0(z)|= 1}of all elements g∈I. The polyhedron Qlooks like an infinite cone in the half space with vertices at 0 and∞whose facets are isometric hyperbolic hexagons paired by hyperbolic isometries from the set I. The group Γ acts on the boundary of Qas a group of automorphisms. Intersection of Qwith the dihedral angle of magnitude 2π/m (which is fundamental polyhedron for the finite cyclic group < K >) is a fundamental polyhedron of the doubly generated group G∗. This shows that G∗is a geometrically infinite normal subgroup of the co-compact hyperbolic group G, with infinite Abelian quotient G/G∗= Γ ∼ =Z⊕Znsatisfying the exact sequence (1.1) and with the limit set Λ(G∗) = Λ(G) = ∂H3 R=C∪ {∞}. Due to Selberg Lemma [A1], there is a torsion free normal subgroup of finite index N / G which gives a normal subgroup of finite index N∗=N∩ G∗/G∗. For some integer k6= 0, we take C=Tk∈Nand denote N∗(C) =< N∗, C >. Then N∗is a normal subgroup of N∗(C) and N∗(C)/N∗∼ =Z, and we have (1.1) with infinite cyclic Abelian group. In this way one has a closed hyperbolic 3-manifold H3 R/N∗(C) fiber over the circle with the fibers homeomorphic to a closed hyperbolic surface S,π1(S)∼ =N∗. 2.3.2. Infinite index normal subgroups with free Abelian quotients of rank ≥2. Given a geometrically finite group in Hn Fwith parabolic subgroups (in particular a non-uniform lattice), one can find its finite index subgroup G⊂Isom Hn Fsuch that each maximal parabolic subgroup Gp⊂Gis either free Abelian or 2-step nilpotent. This is due to the author’s Theorem 6.1 in [AX1]. In particular, parabolic subgroups Gpin each such geometrically finite group G⊂Isom Hn Rin the real hyperbolic space Hn Rare free Abelian of some rank k, with precisely Gp-invariant horoballs Bp⊂Hn Rcentered at the parabolic fixed point p∈∂Hn R. Let P⊂Hn Rbe a fundamental polyhedron of Ghaving the vertex pat infinity. Then Gpacts by automorphisms on the boundary of the polyhedron P∗=∪g∈Gpg(P) whose sides are pairwise identified by elements g∈G\Gp. There are several cases when G\Gpis a subgroup of G, see [A1]. Such subgroups can be taken as normal subgroups G∗⊂Gin (1.1), with the free Abelian quotient G/G∗=Gpand with the limit set Λ(G∗) = Λ(G). In the case of a non-uniform lattice G⊂Isom Hn F, this limit set is the whole sphere at infinity ∂∞Hn F. 2.3.4. Groups of hyperbolic homology 4-cobordisms with wild spherical limit set. Let G⊂Isom H4 Rbe a convex co-compact group which uniformizes a non-trivial homology cobordism, that is a triple (M;N0, N1) where Mis 8 BORIS APANASOV a compact manifold whose interior has a complete hyperbolic structure, int M=H4 R/G, and whose boundary ∂M consists of two disjoint nonhomeomorphic connected closed 3-manifolds N0and N1,∂M =N0∪N1, and the relative homology groups are trivial, H∗(M, N0) = H∗(M, N1) = 0. The discontinuity set Ω(G)⊂∂H4 R=S3of Ghas two invariant components Ω0(a wildly embedded 3-ball in S3) and Ω1=S3\Ω0, and Ni= Ωi/G. The common boundary of these components is the limit set Λ(G) which is homeomorphic to 2-sphere S2and is wildly knotted on a dense subset. Such groups and their non-trivial homology cobordisms were discovered by Apanasov-Tetenov [AT], see also [A1, A3]. Each of these groups was constructed as an HNN-extension of a convex co-compact finitely generated reflection group G0⊂Isom H4 Rby a hyperbolic translation hconjugating two Fuchsian subgroups Γ,Γ0⊂G,G=hG0, h :hΓh−1= Γ0i. Moreover, let Σ be the set of hk-images, k∈Z, of the hyperbolic 3-planes which define reflections generating the reflection group G0. Then they bound a convex hyperbolic polyhedron P0⊂H4 Rwhich is a fundamental polyhedron for an infinitely generated reflection group G∗whose generators are reflections with respect to planes H∈Σ. Clearly, the group G∗is a normal subgroup in G, Λ(G∗) = Λ(G), its quotient G/G∗is the cyclic group <h>which acts by automorphisms on the boundary ∂P0, and these groups satisfy the short exact sequence (1.1). Though the constructed groups Gand G∗/ G have finite order elements, the same procedure as in Section 2.3.1 gives torsion free groups with the same properties. 2.3.5. Bowditch-Mess hyperbolic groups. Now we present another exotic 4-dimensional hyperbolic group G∗which is finitely generated normal subgroup of a geometrically finite (convex cocompact) group G⊂Isom H4 Rwith cyclic quotient G/G∗and satisfying the short exact sequence (1.1). Moreover, these groups are subgroups of a cocompact lattice in Isom H4 Rgenerated by reflections in the faces of the regular hyperbolic 120-cell in H4 Rwhose all dihedral angles are π/2. This group G∗is geometrically infinite and is not finitely presented. Also it has another anomaly that its action on the discontinuity set Ω(G∗) = Ω(G) has the quotient manifold Ω(G∗)/G∗of infinite topological type. This construction is due to Bowditch and Mess [BM]; see also [A1] for other constructions of exotic subgroups of geometrically finite groups with similar “non-finiteness”, including such finitely generated subgroups G∗with infinitely many conjugacy classes of Abelian rank two subgroups, as well as finitely generated subgroups G∗which are not groups of type Fn, see Remark 5.52 in [A1]. To define these groups, we start with a regular dodecahedron in H3 R⊂H4 R INTERSECTIONS OF GEOMETRICALLY FINITE GROUPS 9 whose all dihedral angles are π/2. It defines a hyperbolic orbifold which can be described by the following Thurston’s construction. One represents this dodecahedron combinatorically as an Euclidean cube with six additional disjoint edges splitting cube faces into equal rectangles. Identifying the opposite faces of the cube so as to form a 3-torus, we convert these additional edges into three disjoint embedded circles. Our hyperbolic orbifold is defined by assigning to each of these circles a transverse cone angle equal to π. It is important to note that this orbifold fibres over the circle (given by the long diagonal of the cube). Applying the Selberg Lemma, we obtain a closed hyperbolic 3-manifold fibering over the circle and tiled by dodecahedra. It contains an immersed totally geodesic surface formed as a union of pentagonal faces. Considering a finite covering we may assume that we have such a closed hyperbolic manifold M0with totally geodesic 2-side surface S0(with a wide uniform regular neighborhood in M0). Taking the infinite cyclic covering Mof M0corresponding to the fibre subgroup of π1(M0), we obtain a component Sof the inverse image of S0under the covering projection. This surface S⊂Mis totally geodesic, has infinite topological type, and has the uniform regular neighborhood in M. Now for I= [−1,1] and π1(M)∼ =Γ⊂Isom H3 RIsom H4 R, we realize the product M×Ias a hyperbolic 4-manifold with convex boundary such that M× {0}is universally covered by a hyperbolic 3-space H3 R⊂H4 Rwhose universal regular neighborhood is Γ-invariant. Take another such copy of M×Iand superimpose these two copies such that the two copies of M×{0} sitting inside overlap orthogonally along the surface S. Identifying the superimposed pieces of M×I(after a rotation of the square, I×I, through an angle of π/2), we arrive with a hyperbolic 4-manifold with boundary. It is possible to make proper choices of regular neighborhoods to smooth out the boundary locally and make it convex. It is possible to embed the 4-manifold thus obtained, as a deformation retract, inside a complete hyperbolic 4-manifold without boundary, M4. So we have that the group of covering transformation of M4is our desired group G∗∼ =Γ∗π1(S)Γ acting by isometries in H4 R. The cyclic extension of this group corresponding to fibering over the circle is our geometrically finite group Gcontaining G∗. The fact that this finitely generated group G∗is not finitely presented follows from a result of B.H. Neumann [N]. It says that if the amalgamated free product A∗CBof finitely generated groups Aand Bis finitely presented, then Cis finitely generated. 3. The basic technique for constructing examples Here we will describe the basic twisting technique for creating desired intersections of two geometrically finite groups of isometries of a symmetric