TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS. 1. Introduction

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1 TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI 1. Introduction The theory of hyperbolic groups was introduced and developed by Gromov in the 1980s ([10], [11] and [12]), which makes precise the notion of a negatively-curved group. The basic idea (of geometric group theory in general) is to turn a finitely-generated group into a geometric object and study its geometry, usually making reference to its Cayley graph. Hyperbolic groups are ubiquitous. Many groups arising in classical geometric context such as fundamental groups of compact Riemannian manifolds with strictly negative curvature are hyperbolic. Moreover, most finitely presented groups (in certain probabilistic sense) are also hyperbolic. Hyperbolic groups possess many good algorithmetic and combinatorial properties. In 1911, Dehn proposed the word problem, the conjugacy problem and the isomorphism problem for a class of finitely presented groups. It turns out that hyperbolic groups are a class of groups having solvable word problem and conjugacy problem. Furthermore, in any torsion-free hyperbolic group, the isomorphism problem is also solvable. Hyperbolic groups also have many connections to 3-manifolds and hyperbolic geometry. A positive solution to the famous conjecture of Cannon [6] on the boundary of a hyperbolic group would offer a new approach of tackling the Thurston Hyperbolization Conjecture. Dani Wise, together with collaborators developed a sophisticated theory of special cube complexes and proved (with Bergeron in [4]) that for any closed hyperbolic 3-manifold, there is a non-positively curved compact cube complex with the same fundamental group. Ian Agol completed Wise s program and finally settled the Virtual Haken Conjecture ([1]), which fulfilled Thurston s vision on 3-manifold theory. In this proposal, we explore some basic properties of hyperbolic groups and three-dimensional topology and geometry. 2. Coarse geometry We begin our discussion of hyperbolic groups with some preliminary definitions. Definition 2.1. A metric space X is a path metric space if x, y X, the distance d X (x, y) is equal to inf {L γ : [0, L] X, 1 Lipschitz (i.e. short) such that 0 x, L y}. A metric space X is a geodesic metric space if it is a path metric space and if the infimum is achieved by some γ for all x, y X. Such γ is called a geodesic. A metric space X is proper if x X, d X (x, ) : X R is proper. Therefore, by making each edge isometric to the Euclidean unit interval, the Cayley graph C S (G) of a group G with a fixed generating set S is a geodesic metric space. We denote the induced word metric by S, i.e. g S = d(id, g). Furthermore, the metric space C S (G) is proper if S is finite. Date: February 4,

2 2 YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI Example 2.2. Let G = Z. Consider two generating sets S = {1} and S = {1, 2}. Then C S (G) is a regular graph of degree 2 and we define d(x, y) = x y. On the other hand, for C S (G), define { x y d (x, y) = 2, x y is even x y +1 2, x y is odd Even though two Cayley graphs look very different on small scale, the metrics d and d, on large scale, differ only by a multiplicative and an additive constant. This suggests that we need to coarsen the geometric category. Definition 2.3. Let (X, d X ) and (Y, d Y ) be metric spaces. A map f : X Y is called a quasiisometric map (embedding) if there exist constants K 1, ε 0 such that K 1 d X (x, y) ε d Y (f(x), f(y)) Kd X (x, y) + ε. f is a quasi-isometry (QI) if y Y, x such that d Y (f(x), y) ε. Remark 2.4. In coarse geometry, we only care about maps up to a bounded distance. Lemma 2.5. Any two Cayley graphs of a finitely generated group are QI. Proof. (Sketch) The identity map from the group to itself with respect to two finite generating sets is a quasi-isometry. The following are some basic properties of quasi-isometries, from which we see that QI is an equivalence relation on metric spaces. Proposition 2.6. (Basic properties of QI) (1) If f : X Y and g : Y Z are QI, then so is g f. (2) If f : X Y is a QI, then there is a QI g : Y X such that f g and g f are bounded distance from the identity maps. Theorem 2.7. (Milnor-Schwarz) If G acts geometrically on a proper geodesic metric space X, then G is finitely generated and for any finite generating set, the Cayley graph of G is QI to X. Proof. (Sketch) Our proof follows [5]. Let B = B 2R+1 (x) such that B R (x) projects onto X/G. Since the action of G is properly discontinuous, there are only finitely many nontrivial elements g i G such that B g i B. Let S be the subset of G consisting of these elements. We claim that S is a generating set of G. Let g G and let γ be the shortest geodesic connecting x and gx. Parametrizing γ by arclength and for each integer i (0, γ ), let g i be such that d X (g i x, γ(i)) R. Then B g 1 j g j+1 B. Hence, g j+1 = g j s i(j), i.e. S generates G. Moreover, g S γ + 1 = d(x, gx) + 1. Conversely, d(x, gx) (4R + 2) g S, since for each g i g 1 i+1 S, the length of the path connecting g ix to g i+1 x is at most 4R + 2. Hence, the geometric structure associated to a group is defined up to quasi-isometry. 3. Hyperbolic spaces and quasigeodesics The main goal of this section is to show that hyperbolicity (in the sense of Gromov) is invariant under quasi-isometries, which, together with Lemma 2.5, will allow us to define the key object of interest hyperbolic groups. Definition 3.1. A geodesic metric space (X, d X ) is δ-hyperbolic if for any geodesic triangle, each side is contained in the δ-neighbourhood of the union of the other two sides. A metric space is hyperbolic if it is δ-hyperbolic for some δ 0.

3 TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS 3 Examples 3.2. (1) (R-)trees are 0-hyperbolic. (2) H n is (log 2 + 1)-hyperbolic. (3) R n is not hyperbolic. The next lemma states that there is a notion of projection from a hyperbolic space to any geodesics. Lemma 3.3. Let X be δ-hyperbolic, let γ be a geodesic segment/ray/line in X, and let p X. Then there is a point q on γ realizing the infimum of distance from p to points on γ, and moreover for any two such points q and q, we have d X (q, q ) 4δ. Remark 3.4. Besides this nearest point construction, there are other equivalent constructions using geodesic triangles and Gromov product. Interested readers are referred to [3] (Section 6.3) for details. Quasigeodesic is another fundamental concept in geometric group theory as quasi-isometries do not preserve geodesity. In fact, as we will see in Lemma 3.8, quasigeodesics are not too far away from being geodesics. Definition 3.5. A (K, ɛ)-quasigeodesic segment/ray/line is the image of a segment/ray/line in R under a (K, ɛ)-qi map. Definition 3.6. Suppose P, Q are subsets of a metric space (X, d). The Hausdorff distance between P and Q is the infimum of those r [0, ] for which P N(Q, r) and Q N(P, r) Remark 3.7. This is a pseudometric on the set of all bounded subsets of X. It is a metric when restricted to the set of closed subsets of X. Now we show that hyperbolicity is preserved under quasi-isometries. Lemma 3.8. (Morse lemma) Let (X, d X ) be a proper δ-hyperbolic space. Then for any (K, ε) there is a constant C K,ε,δ such that any (K, ε)-quasigeodesic γ is within Hausdorff distance C of a genuine geodesic γ g. If γ has one or two endpoints, γ g can be chosen to have the same endpoints. Proof. (Sketch) Our proof follows [5]. It suffices to prove the lemma when γ is a finite segment. We need to estimate the Hausdorff distance between γ and a geodesic γ g having the same endpoints. Fix a constant C and suppose p, p γ are such that d(p, γ g ) = d(p, γ g ) = C and d(x, γ g ) C, x γ. Choose p i γ in such a way that their projections q i γ g satisfy d(q i, q i+1 ) = 11δ. This guarantees that there are points r i p i p i+1 and s i q i q i+1 such that d(r i, s i ) 2δ. Hence, d(p i, p i+1 ) 2C 4δ and the length of γ between p and p is at least (2C 4δ)d(q, q )/11δ, where q, q γ g are projections of p and p. On the other hand, we also have d(p, p ) 2C + d(q, q ). Hence, if d(q, q ) is sufficiently large, there is a uniform bound for C depending on K, ε and δ. Corollary 3.9. (QI-invariance of hyperbolicity) Let Y be δ-hyperbolic and let f : X Y be a (K, ε)-qi. Then X is δ -hyperbolic for some δ. Proof. (Sketch) Let a, b and c be the vertices of a geodesic triangle in X. Then f(ab) is within distance 2C + δ of f(ac) f(bc). Hence, ab is within distance K(2C + δ) + ε of ac bc. 4. Hyperbolic groups In this section, we define hyperbolic groups and study their subgroups and finiteness properties. Definition 4.1. A finitely-generated group G is hyperbolic if for some (and hence for all) finite generating set S, the Cayley graph C S (G) is δ-hyperbolic for some δ 0. Remark 4.2. In his original paper [11], Gromov gave three equivalent definitions of hyperbolic groups in terms of isoperimetric inequalities, the Gromov product and tangent subcones at infinity. In the next section, we will show that linear isoperimetric inequality characterizes hyperbolicity.

4 4 YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI An immediate consequence of Milnor-Schwarz lemma and Corollary 3.9 is : Proposition 4.3. If G acts geometrically on a proper hyperbolic space, then G is hyperbolic. Examples 4.4. (1) Any finite group. (2) Any virtually free group. (3) π 1 of any compact hyperbolic manifold. (4) π 1 (Σ g ), g 2. Non-examples 4.5. ([3]) (1) Z n for any n 2. (2) A hyperbolic group cannot contain any Z 2 subgroup (see Proposition 4.6). Hence, SL(n, Z), mapping class groups and knot groups cannot be hyperbolic Subgroups of hyperbolic groups. The main goal of this subsection is to prove Proposition 4.6. A hyperbolic group G cannot contain any Z 2 subgroup. (also see [5], Lemma and Lemma 2.3.7) Proof. (Sketch) Suppose g G has infinite order so that < g > Z. Then the bi-infinite path γ = n Z [gn x, g n+1 x] is a quasigeodesic for any vertex x in the Cayley graph. Equivalently, the translation length of g, τ(g) := lim n d(x, g n x)/n, is positive. Let h G such that gh = hg. Consider the bi-infinite path hγ = n Z [gn hx, g n+1 hx]. By Morse lemma, the Hausdorff distance between γ and hγ depends only on the quasi-geodesic constants of γ. However, there are only finitely many choices for hγ since G is finitely generated. Therefore, G cannot contain any Z 2 subgroup. Remark 4.7. A similar argument shows that a hyperbolic group cannot contain any subgroup isomorphic to a Baumslag-Solitar group which is of the form B(p, q) =< a, b a p b = ba q >, where p, q 1 (note that Z 2 = B(1, 1)) Finiteness of hyperbolic groups. By definition, hyperbolic groups are finitely generated. Now, we show that hyperbolic groups are also finitely presented. The key ingredient is the Rips complex. Theorem 4.8. (Rips) Let G be a hyperbolic group. Then there exists a contractible simplicial complex K on which G acts properly discontinuously and cocompactly (p.d.c.). Proof. (1) (Constructing Rips complex, [11] p.95) Let X be the Cayley graph of G. Let d > 0 and consider the simplicial polyhedron P d (X) whose vertices are the same as those in X and where a finite subset Y X spans a simplex in P d (X) if and only if d(p, q) d, p, q Y. Lemma 4.9. Let X be a δ-hyperbolic space. If there exists ε > 0 such that x X with x 1 2 d, there exists a point x X with x min( x ε, x 2δ) and x x d 2δ, then P d (X) is contractible. (2) ([11] p.101) The action of G on P d (X) is obviously p.d.c. Take ε = 1 and d 4δ + 1, by Lemma 4.9, P d (X) is contractible. Corollary Hyperbolic groups are finitely presented.

5 TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS 5 5. The linear isoperimetric inequality and hyperbolicity The main purpose of this section is to show that the linear isoperimetric inequality characterizes hyperbolicity of groups. We begin with some definitions. Definitions 5.1. (1) Let G =< S R > be a finitely presented group. Let w be a word in S such that w = e in G. The area of w is defined to be A(w) = min {n w = n i=1 u ir ±1 i u 1 i }, where u i (the free group) F (S) and r i R. (2) A function f : [0, ) [0, ) is an isoperimetric function for G if for all w F (S) with w = e in G, we have A(w) f( w ). We say that G satisfies a linear/quadratic/exponential isoperimetric inequality if the isoperimetric function is linear/quadratic/exponential. (3) Let f and g be two isoperimetric functions. We say f g if there exists a constant K, K > 0 such that f(x) Kg(Kx + K) + Kx + K and g(x) K f(k x + K ) + K x + K. Theorem 5.2. Hyperbolic groups satisfy a linear isoperimetric inequality. Proof. Without loss of generality, we can assume that the Cayley graph is δ-hyperbolic, for some integer δ > 0. We will prove by induction on w that A(w) K w, for all w = e in G, where K = max{a(w) w 10δ}. Suppose it is true for w < n, where n 10δ. When w = n + 1, we have three cases. Case 1: All vertices w(i) of w are in N 5δ (e). Then take a shortest path joining e and w(5δ). Case 2: If Case 1 is not true, let w(t) be the furthest vertex of w from e. If d(w(t), w(t ± 5δ)) < 5δ, then apply Case 1. Case 3: Otherwise, consider the geodesic triangles with vertices, e, w(t 5δ), w(t) and e, w(t + 5δ), w(t). Then d(w(t 2δ), w(t + 2δ)) 2δ and apply Case 1 and 2. Now, we prove the converse. Theorem 5.3. If a finitely generated group G satisfies a linear isoperimetric inequality, then it is hyperbolic. Proof. Suppose G is not hyperbolic. Then in the cayley graph there exists a geodesic triangle abc and a point on it which is at least distance δ away from the union of the other two sides. Truncate the triangle into a hexagon whose vertices are δ 10 away from a, b and c. Let l 1, l 2 and l 3 be the length of the truncated sides and let l = max{l 1, l 2, l 3 }, which is realized by a geodesic L. Since l( H) < 6l, A(H) < 6Kl for some constant K as G satisfies a linear isoperimetric inequality. Let ρ be the maximum length of a relator of G. For Star(L), the star neighbourhood of L, we have A(Star(L)) l ρ and l( (Star(L)) H) l 2ρ. Repeating 12K times, we get A(Star 12K (L)) l ρ + l 2ρ ρ contradiction l 12Kρ ρ > 6Kl, which is a Remark 5.4. In fact, if a group satisfies a subquadratic isoperimetric inequality (i.e. lim x f(x) x 2 = 0), then it satisfies a linear one. Examples 5.5. (1) The Heisenberg group satisfies a cubic isoperimetric inequality. (2) Z 2 satisfies a quadratic isoperimetric inequality, but not a linear one. (3) CAT (0) spaces satisfy a quadratic isoperimetric inequality. 6. The Gromov boundary This section follows [14] (section 2) and [5]. The ordinary hyperbolic n-space H n has a natural asymptotic boundary which is topologically an (n 1)-sphere and can be thought of as a natural compactification of the space. There is an analogous construction for hyperbolic groups (or more generally geodesic metric spaces). In this section, we describe the construction of the Gromov

6 6 YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI boundary of hyperbolic spaces and show that it is compact and metrizable. More details about the boundary of hyperbolic groups can be found in the survey paper [14] by Kampovich and Benakli. Definition 6.1. (Geodesic boundary) Let X be a hyperbolic space. The geodesic boundary of X is defined to be g X := {[γ] γ : [0, ) X is a geodesic ray in X}, where two geodesics are equivalent if they are a finite Hausdorff distance apart. Definition 6.2. (Sequential boundary) Let X be a hyperbolic space and x X be a base point. A sequence of points (x n ) n 1 in X is said to converge at infinity if lim inf i,j (x i, x j ) x =. Two sequences (x n ) n 1 and (y n ) n 1 are equivalent if lim inf i,j (x i, y j ) x =. We define the sequential boundary of X to be X := {[(x n ) n 1 ] (x n ) n 1 is a sequence converging at infinity in X}. Remark 6.3. The above two definitions coincide if the metric space is proper, since the map f : g X X given by f([γ]) = [(γ(n)) n 1 ] is a bijection. Similar to H n, the boundary of a hyperbolic space carries a natural topology. Definition 6.4. (Topology on g X) Let X be a hyperbolic space and x X be a base point. For any p g X, define the basis of neighbourhoods to be {V (p, r) r 0}, where V (p, r) := {q g X for some geodesic rays γ 1, γ 2 with [γ 1 ] = p, [γ 2 ] = q, lim inf t (γ 1 (t), γ 2 (t)) x r}. Definition 6.5. (Topology on X) Let X be a hyperbolic space and x X be a base point. For any p X, define the basis of neighbourhoods to be {U(p, r) r 0}, where U(p, r) := {q X for some sequences (x n ) n 1, (y n ) n 1 with [(x n ) n 1 ] = p, [(y n ) n 1 ] = q, lim inf i,j (x i, y j ) x r}. This also gives rise to a topology on X X by setting the basis of a point p X to be V (p, r) := V (p, r) {y X for some geodesic γ with [γ] = p, lim inf t (γ(t), y) x r} Remark 6.6. (1) Again, when the metric space is proper, two topologies coincide. (2) For a proper hyperbolic space X, the topological space X is compact. Furthermore, the boundary of a proper hyperbolic space is in fact metrizable. Definition 6.7. Let X be a proper metric space. Let a > 1 and x 0 X be a base point. A metric d a on X is a visual metric with respect to a and x 0 if there exists a constant C > 0 such that d a induces the same topology on X and for any two points p, q X, we have C 1 a d(x0,γ) d a (p, q) Ca d(x0,γ), where γ is a bi-infinite geodesic connecting p and q. Proposition 6.8. Let X be a proper hyperbolic space with base point x 0. Then there exists a 0 > 1 such that for any a (1, a 0 ), the boundary X admits a visual metric d a with respect to a and x 0. Proof. (Sketch) Consider the metric defined by d a (p, q) := inf γ γ a x, where the infimum is taken over continuous paths γ joining p and q. Examples 6.9. (1) The boundary of a compact space is empty. Thus the boundary of any finite group is empty. (2) The real line has two boundary points. The boundary of group Z or any virtually cyclic group, is thus the two-point space.

7 TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS 7 (3) If p 3, then T p is a Cantor set. Thus F n is a Cantor set for all n 2. Example (Convergence group action) A hyperbolic group G acts by homeomorphisms on G. It turns out that the action has a dynamical property, namely, it is a uniform convergence action. A famous theorem of Bowditch proves the converse. See [5] Lemma and Theorem for details. One of the important results obtained by using the convergence group action is the characterization of hyperbolic groups with circle boundary. This is a theorem due to Tukia, Gabai, Freden, Casson and Jungreis which states that if the boundary of a hyperbolic group is homeomorphic to S 1, then the group is virtually fuchsian. See, for example, [9]. Example (Tits alternative for hyperbolic groups) Let G be a hyperbolic group and let g 1 and g 2 be non-torsion elements of G. Then either < g 1, g 2 > is virtually Z or there exists a natural number n > 0 such that < g n 1, g n 2 > is isomorphic to the free group on two generators. To see this, let a ± 1 and a± 2 be the fixed points of g 1 and g 2 respectively. If they are not all distinct, then the subgroup generated by g 1 and g 2 is virtually Z. For the stabilizers of a point on the boundary is virtually Z. Otherwise, there are four pairwise disjoint neighbourhoods of fixed points so that we can apply the ping-pong lemma to obtain the conclusion. 7. Loop and Sphere Theorems The rest of the proposal is devoted to the study of three-dimensional topology and geometry. We start with basic homotopy properties of 3-manifolds The Loop Theorem. Theorem 7.1. (Dehn s Lemma). Let M be a 3-manifold with boundary. If f : (D 2, S 1 ) (M, M) is an embedding on some neighbourhood of S 1, then it extends to an embedding f : (D 2, S 1 ) (M, M). Dehn s lemma was first rigorously proved by Papakyriakopoulos [15], together with another two foundational results, i.e. the loop theorem and the sphere theorem. The following version of the loop theorem was formulated by Stallings in [19], from which we can easily deduce Dehn s lemma. Theorem 7.2. (Loop Theorem). Let M be a 3-manifold and S M be compact. Let N be a normal subgroup of π 1 (S) which does not contain the kernel of the map π 1 (S) π 1 (M). Then there exists a proper embedding f : (D 2, S 1 ) (M, S) such that the conjugacy class of f(s 1 ) is not contained in N. Proof. (Sketch) Replace f by a simplicial map f 0 and construct a tower. Show that the tower terminates at some finite step and the boundary of some tubular neighbourhood at the top level consists of spheres. Obtain an embedding g n : (D 2, S 1 ) (M n, S n ). Descend down the tower and modify g n to eliminate singularities. Proof. (of Dehn s Lemma) Let S be a neighbourhood of f(s 1 ) in M and let N = {e}. By the loop theorem, there is an embedding f : (D 2, S 1 ) (M, S) with f(s 1 ) nontrivial. Hence, up to isotopy, f = f on the neighbourhood of S 1. Another important application of the Loop Theorem is the following Corollary 7.3. (Kneser s Lemma) Let S M be a 2-sided embedded surface which is not π 1 - injective. Then there is a properly embedded disk in M whose boundary is nontrivial in S. Proof. Let γ be a loop that is nontrivial in S but trivial in M so that it bounds a disk f : D M. Perturb f if necessary so that it is transverse to S. Since S is embedded and 2-sided, D S is a set of disjoint circles. Eliminate those circles, from the innermost one, to obtain a disk D whose interior maps to M S and whose boundary is nontrivial in S. Apply the loop theorem to M S.

8 8 YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI 7.2. The Sphere Theorem. Theorem 7.4. (Sphere Theorem). Let M be a compact 3-manifold with π 2 (M) nontrivial. Then there exists a 2-sided embedded S 2 or RP 2 representing a nontrivial element in π 2 (M). Remark 7.5. A 2-sided embedded RP 2 exists only in a nonorientable 3-manifold. On the other hand, as we will see in the following example, RP 2 is also necessary for the nonorientable case. Example 7.6. Let M = RP 2 S 1. Then M is compact and has nontrivial π 2 M. However, every embedded S 2 is trivial in π 2 M since M is irreducible. Proof. (Sketch) Let p : M M be the universal cover. It is a fact that we can represent a nontrivial class in π 2 (M) = π 2 ( M) = H 2 ( M) by a closed embedded orientable surface S M with each boundary component being π 1 -injective. Since M is simply connected, each boundary component of S is a sphere. Call a nontrivial one of these spheres S 0. To prove the theorem, it suffices to prove that all the translations τ(s 0 ), where τ π 1 M, are disjoint embedded spheres so that the restriction p S0 is a covering map. We prove this by using normal form arguments. Pick a triangulation of M and lift it to the triangulation T of M. We may take all the translates τ(s 0 ) to be in normal form with minimal weight, having transverse intersections disjoint from the 1-skeleton of T. Then we modify the spheres to be disjoint in all 2 and 3-simplices and show that the intersection of all the spheres with 2-simplices and spheres themselves are invariant under the deck group action. See [13] for details. 8. Prime Decomposition For an orientable 3-manifold M with nontrivial π 2 M, the Sphere Theorem asserts that M contains an embedded separating 2-sphere. Splitting M along this sphere and filling the boundary with balls produces two manifolds M 1 and M 2 such that M = M 1 #M 2. We say that M is prime if M = M 1 #M 2 implies that M 1 = S 3 or M 2 = S 3. In other words, the splitting sphere is the boundary of a ball in M. M is said to be irreducible if every embedded S 2 M bounds a ball B 3 M. Theorem 8.1. (Alenxander) Every smoothly embedded S 2 in S 3 bounds a ball. Proof. (Sketch) Let h : S 2 R be the height function such that all its critical values are isolated. Let a 1 <... < a n be noncritical values of h such that each interval (a i, a i+1 ) contains precisely one critical value. Then h 1 (a i ) consists of disjoint circles. Surger S 2 along these circles, starting from the innermost one. After the surgery, the original sphere becomes a disjoint union of spheres. There are seven isotopy types for these spheres, each of which bounds a ball. Reverse the surgery process and at each stage we have a collection of spheres, each bounding a ball. Remark 8.2. Alexander s theorem is essential to make sense of the prime decomposition as it implies that S 3 is prime. Example 8.3. S 1 S 2 is the only orientable prime but not irreducible 3-manifold. For if M 3 is prime but not irreducible, then there exists a nonseparating S 2. Consider a neighbourhood N of S 2 I. Then N is diffeomorphic to S 1 S 2 minus a ball. Now we state the Prime Decomposition Theorem due to Kneser and Milnor. Theorem 8.4. (Prime Decomposition) Let M be a compact, connected and orientable 3-manifold. Then there exists a decomposition of M = P 1 #...#P n, where each P i is prime. Moreover, this decomposition is unique up to permutations of P i s. Remark 8.5. The Prime Decomposition corresponds to the fact in group theory that a finitely generated group cannot be split as a free product of infinitely many nontrivial factors.

9 TOPIC PROPOSAL HYPERBOLIC GROUPS AND 3-MANIFOLDS 9 9. Torus Decomposition The prime decomposition reduces the study of 3-manifolds to the study of irreducible ones. The next step is to consider decomposing an irreducible 3-manifold by splitting along embedded tori. Definition 9.1. A 2-sided surface S M 3 without S 2 or D 2 components is incompressible if for each disk D M with D S = D there is a disk D S such that D = D. The disk D is called a compressing disk. An irreducible manifold M is atoroidal if every incompressible torus in M is isotpic, relative to its boundary, to a subsurface of M. We state without proof the existence of Torus Decomposition. Proposition 9.2. Let M be a compact connected irreducible 3-manifold. Then there exists a finite collection T of disjoint incompressible tori such that each component of M T is atoroidal. This decomposition is not unique as demonstrated in the following example. Example 9.3. ([13]) For i = 1, 2, 3, 4, let M i be a solid torus with boundary decomposed into two annuli A i and A i, each of which winds the S1 factor of M q i > 1 times. Glue each A i to A i+1 and call the resulting manifold M. Let T 1 = A 1 A 3 and T 2 = A 2 A 4. Then, if q i s are pairwise coprime, M T 1 and M T 2 are not homeomorphic as they have different fundamental groups. The manifold in the above example belongs to a special class of manifolds which was extensively studied by Seifert in 1930s. We define it now. Definition 9.4. A model Seifert fibring of S 1 D 2 is a S 1 -bundle obtained by identifying the disks {0} D 2 with {1} D 2 via a 2πp/q rotation, where (p, q) = 1. A Seifert fibring of a 3-manifold M is S 1 -bundle where each fibre has a neighbourhood diffeomorphic to a neighbourhood of a fibre in some model Seifert fibring. A Seifert manifold is a manifold with a Seifert fibring. The crucial fact is that Seifert manifolds are the only exceptions for the uniqueness of torus decomposition. Theorem 9.5. (Jaco-Shalen, Johannson) For a compact irreducible orientable 3-manifold M, there is a collection T of disjoint incompressible tori such that each component of M T is either atoroidal or a Seifert manifold, and a minimal such collection T is unique up to isotopy. Example 9.6. Let M 3 be a surface bundle over S 1 with fibre Σ g, where g 2. Such a bundle is in one-to-one correspondence with the mapping class group Mod(Σ g ). If φ Mod(Σ g ) is finite order, then M is Seifert fibred. If φ is pseudo-anosov, then M is atoroidal and not Seifert fibred. 10. Three-dimensional geometries A manifold has a geometric structure if it is locally isometric to some homogeneous space. More precisely, Thurston gives the following definition. Definition A model geometry (G, X) is a manifold X together with a Lie group G of diffeomorphisms of X such that (a) X is connected and simply connected; (b) G acts transitively on X with point stabilizers being compact; (c) G is not contained in any larger group of diffeomorphisms of X with compact point stabilizers; (d) there exists at least one compact manifold modelle on (G, X). A manifold is said to be modelled on (G, X) if it is diffeomorphic to X/Γ for some discrete subgroup Γ of G which acts freely on X.

10 10 YAN MARY HE DISCUSSED WITH PROFESSOR DANNY CALEGARI The classification of model geometries in dimension two is a consequence of the Uniformization Theorem. These are spaces of constant curvature, namely, S 2, E 2 and H 2 with their respective isometry groups. Although their three-dimensional analogues are also model geometries, the complete list of three-dimensional geometries was not known until Thurston s work in the 1970s. Theorem ([20]) There are eight three-dimensional model geometries (G, X): (a) S 3, E 3 or H 3, if the point stabilizers are three-dimensional. (b) If the point stabilizers are one-dimensional, (b1) S 2 E 1 or H 2 E 1, if the G-invariant Riemannian metric has curvature 0. (b2) Nil or SL(2, R), if the curvature is 1. (c)sol, if the point stabilizers are zero-dimensional. Proof. Let G be the identity component of G which acts with stabilizers G x of points x X. G x is a connected subgroup of SO(3). (a) If G x = SO(3), then the X has constant curvature. Therefore, X = S 3, E 3 or H 3. (b) If G x = SO(2), then there is a G -invariant vector field V given by the eigenspace whose trajectories define a one-dimensional foliation F on X. Since each leaf is invariant under SO(2), it is S 1 or R and hence Y = X/F is one of E 2, S 2 or H 2. The plane field orthogonal to F is a connection for the bundle. (b1) If the curvature is zero, then the bundle is trivial and X = S 2 E 1 or X = H 2 E 1. (b2) If the curvature is not zero, then the plane field defines a contact structure. Y = S 2 does not give a geometry. Y = E 2 gives nilgeometry. If Y = H 2, then X = SL(2, R). (c) If G x is trivial, then X is a Lie group and the classification is done by studying its Lie algebra. Examples (a) T 3 admits an Euclidean structure. Lens spaces are spherical manifolds. Hyperbolic 3-manifolds, of course, admit H 3 -structures. (b1) There are only four compact manifolds admitting an S 2 E 1 -structure: S 2 S 1, S 2 S 1, RP 2 S 1 and RP 3 #RP 3. There are infinitely many compact manifolds with an H 2 E 1 -structure: Σ g 2 R. (b2) Nil geometry: Heisenberg group. SL(2, R) geometry: unit tangent bundle of a hyperbolic surface. (c) Sol geometry: T 2 -bundle over S 1 with Anosov monodromy. It is not true in general that any 3-manifold admits a geometric structure. However, Thurston conjectured in the 1970s and now is the geometrization theorem that every orientable closed 3- manifold can be split along spheres and tori so that each piece has a complete geometric structure with finite volume. If a closed 3-manifold is simply connected, then it does not have a canonical decomposition and therefore must admit a geometric structure. Hence, Thurston geometrization implies the famous Poincaré conjecture proved by Perelman([16], [17] and [18]) that every closed 3-manifold with trivial fundamental group is homeomorphic to S 3. References [1] I. Agol, The virtual Haken conjecture. arxiv: v1 [2] J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on hyperbolic groups, Group Theory from a Geometrical Viewpoint, Proceedings of the workshop held in Trieste, World Scientific, Singapore (1991). [3] B. Bowditch, A course on geometric group theory. MSJ Memoirs, 16. Mathematical Society of Japan, Tokyo, [4] N. Bergeron and D. Wise, A boundary criterion for cubulation, Amer. J. Math 134 (2012), no.3, [5] D. Calegari, The ergodic theory of hyperbolic groups, Contemp. Math. to appear. [6] J. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics and hyperbolic spaces, Oxford Univ. Press, New York, 1991, pp [7] M. Coornaert, T. Delzant, and A. Papadopoulos, Geometrie et theorie des groupes Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, [8] M. Dehn, Papers on group theory and topology, translated and introduced by J. Stillwell, Springer-Verlag, 1987.

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