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The Coarse Geometry of Groups Tim Susse CUNY Graduate Center December 2, 2011 Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 1 / 38 Outline 1 Groups as Geometric Objects 2 Quasi-isometries 3 Hyperbolicity and Hyperbolic Groups Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 2 / 38 The Cayley Graph Given a presentation of a group G = hS | Ri we associate a geometric object called the Cayley graph, denoted Cayley(G, S). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 3 / 38 The Cayley Graph Given a presentation of a group G = hS | Ri we associate a geometric object called the Cayley graph, denoted Cayley(G, S). Vertex Set = G g1 ∼ g2 if and only if there exists s ∈ S ∪ S −1 with g1 = g2 s Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 3 / 38 The Cayley Graph Given a presentation of a group G = hS | Ri we associate a geometric object called the Cayley graph, denoted Cayley(G, S). Vertex Set = G g1 ∼ g2 if and only if there exists s ∈ S ∪ S −1 with g1 = g2 s Thus the Cayley graph of a group depends on the choice of generating set. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 3 / 38 Cayley Graph of Z2 Figure: Cayley(Z2 , {(1, 0), (0, 1)}) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 4 / 38 Cayley Graphs of Z/7Z (a) (b) Figure: (a) Cayley(Z/7Z, {1}) (b) Cayley(Z/7Z, {2, 3}) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 5 / 38 Two Cayley Graphs of Z (a) (b) Figure: (a) Cayley(Z, {1}) (b) Cayley(Z, {2, 3}) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 6 / 38 Two Cayley Graphs of Z (a) (b) Figure: (a) Cayley(Z, {1}) (b) Cayley(Z, {2, 3}) You might notice that if you step really far back, or zoom out on the graphic, the two Cayley graphs look very similar. We want to formalize and take advantage of this. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 6 / 38 Word Metric on a Group Given a generating set S for G, we define the distance between two points in G to be their distance in the Cayley graph Cayley(G, S). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 7 / 38 Word Metric on a Group Given a generating set S for G, we define the distance between two points in G to be their distance in the Cayley graph Cayley(G, S). Again, this depends on the choice of generating set. This metric dS is called the word metric on G. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 7 / 38 Word Metric on a Group Given a generating set S for G, we define the distance between two points in G to be their distance in the Cayley graph Cayley(G, S). Again, this depends on the choice of generating set. This metric dS is called the word metric on G. Facts For any g ∈ G, dS (g, e) = lS (g), the word length of G. For any g, h ∈ G, dS (g, h) = lS (g −1 h). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 7 / 38 Word Metric on a Group Given a generating set S for G, we define the distance between two points in G to be their distance in the Cayley graph Cayley(G, S). Again, this depends on the choice of generating set. This metric dS is called the word metric on G. Facts For any g ∈ G, dS (g, e) = lS (g), the word length of G. For any g, h ∈ G, dS (g, h) = lS (g −1 h). G acts on (G, dS ) by left multiplication, an isometry. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 7 / 38 Word Metric on a Group Given a generating set S for G, we define the distance between two points in G to be their distance in the Cayley graph Cayley(G, S). Again, this depends on the choice of generating set. This metric dS is called the word metric on G. Facts For any g ∈ G, dS (g, e) = lS (g), the word length of G. For any g, h ∈ G, dS (g, h) = lS (g −1 h). G acts on (G, dS ) by left multiplication, an isometry. If S is a finite generating set, closed balls are finite, so (G, dS ) is a proper metric space. The converse is also true. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 7 / 38 Outline 1 Groups as Geometric Objects 2 Quasi-isometries 3 Hyperbolicity and Hyperbolic Groups Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 8 / 38 Bilipschitz Equivalence Given two metric spaces (X , dX ) and (Y , dY ) we say that a map f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we have: 1 dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ). k Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 9 / 38 Bilipschitz Equivalence Given two metric spaces (X , dX ) and (Y , dY ) we say that a map f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we have: 1 dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ). k A bilipschitz map is like a stretching of the metric at every point (by bounded amounts). It is worth noting the a bilipschitz map is always a topological embedding. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 9 / 38 Bilipschitz Equivalence Given two metric spaces (X , dX ) and (Y , dY ) we say that a map f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we have: 1 dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ). k A bilipschitz map is like a stretching of the metric at every point (by bounded amounts). It is worth noting the a bilipschitz map is always a topological embedding. A weaker notion of equivalence is a ”large scale” or coarse bilipschitz condition. We call this a quasi-isometry. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 9 / 38 Quasi-isometric embeddings Definition. We say that a map f : X → Y is a (k , c) quasi-isometric embedding if for every pair x1 , x2 we have that 1 dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ) + c. k Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 10 / 38 Quasi-isometric embeddings Definition. We say that a map f : X → Y is a (k , c) quasi-isometric embedding if for every pair x1 , x2 we have that 1 dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ) + c. k Further, if the map is coarsely onto (i.e. if every point in Y is distance at most c from f (X )), we call it a quasi-isometry. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 10 / 38 Quasi-isometric embeddings Quasi-isometry defines an equivalence relation on metric spaces: Clearly the identity map is an isometry, so any space is quasi-isometric to itself. A quick computation shows that if f and g are quasi-isometries, so is f ◦ g Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 11 / 38 Quasi-isometric embeddings Quasi-isometry defines an equivalence relation on metric spaces: Clearly the identity map is an isometry, so any space is quasi-isometric to itself. A quick computation shows that if f and g are quasi-isometries, so is f ◦ g Symmetry is a little tricky. To show symmetry we need to construct a ”coarse inverse” of a quasi-isometry f . By this we mean a quasi-isometry f −1 so that there exists a constant r with dX (x, f −1 ◦ f (x)) ≤ r . Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 11 / 38 Coarse Inverse of a Quasi-isometry To construct the inverse, we first need some quick facts. If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see this, look at the left-side of the definition of quasi-isometry: 1 dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0. k Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 12 / 38 Coarse Inverse of a Quasi-isometry To construct the inverse, we first need some quick facts. If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see this, look at the left-side of the definition of quasi-isometry: 1 dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0. k So, up to a bounded diameter ”error”, for each y ∈ f (X ), we can choose g(y ) = x, where we choose some element x ∈ f −1 (y). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 12 / 38 Coarse Inverse of a Quasi-isometry To construct the inverse, we first need some quick facts. If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see this, look at the left-side of the definition of quasi-isometry: 1 dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0. k So, up to a bounded diameter ”error”, for each y ∈ f (X ), we can choose g(y ) = x, where we choose some element x ∈ f −1 (y). For each element y 6∈ f (X ), we note that there exists some y 0 ∈ f (X ) with dY (y, y 0 ) ≤ c. Choose some y 0 with this property and let g(y) = x where x ∈ f −1 (y ). This function is well-defined if we ignore these finite diameter ”errors”. We call this sort of thing coarsely well-defined. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 12 / 38 Why is Quasi-isometry Right? Given a finitely generated group G, there exist many finite generating sets. Proposition. Let S and T be two finite generating sets of a group G, then the identity map on G is a quasi-isometry idG : Cayley(G, S) → Cayley(G, T ). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 13 / 38 Why is Quasi-isometry Right? Given a finitely generated group G, there exist many finite generating sets. Proposition. Let S and T be two finite generating sets of a group G, then the identity map on G is a quasi-isometry idG : Cayley(G, S) → Cayley(G, T ). Proof. Let S = {s1 , . . . , sn } and T = {t1 , . . . tm }. Since S generates G there exists a shortest (geodesic) spelling of each ti in the language of 1 S. In particular, ti = s1i · · · skik , where j = ±1. Let K = max{k : lS (ti ) = k}, i.e. the longest word length of a ti in Cayley(G, S). Similarly, let L be the maximum word length of the si in Cayley(G, T ). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 13 / 38 Why is Quasi-isometry Right? Take g ∈ G. Then say lS (g) = r . so g = s1 (g) · · · sr (g), where each si (g) ∈ S ∪ S −1 . Replace each of the si with their spellings in t, and we see that lT (g) ≤ Kr . Similarly, if lT (g) = r , then lS (g) ≤ Lr . So, for any g ∈ G we get: 1 · lT (g) ≤ lS (g) ≤ L · lT (g). K However, for any g, h ∈ G, dS (g, h) = lS (gh−1 ), and similarly for T . So, this turns in to a quasi-isometry. In fact, this is a bi-Lipschitz equivalence. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 14 / 38 Why is Quasi-isometry Right? Take g ∈ G. Then say lS (g) = r . so g = s1 (g) · · · sr (g), where each si (g) ∈ S ∪ S −1 . Replace each of the si with their spellings in t, and we see that lT (g) ≤ Kr . Similarly, if lT (g) = r , then lS (g) ≤ Lr . So, for any g ∈ G we get: 1 · lT (g) ≤ lS (g) ≤ L · lT (g). K However, for any g, h ∈ G, dS (g, h) = lS (gh−1 ), and similarly for T . So, this turns in to a quasi-isometry. In fact, this is a bi-Lipschitz equivalence. Notice in our Cayley graphs for Z before we had K = 3, L = 2 for S = {1} and T = {2, 3}. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 14 / 38 The Milnor-Svarc Lemma One of the fundamental tools in studying the coarse geometry of groups is the following fact. Theorem. Let X be a proper metric space and let G act on X geometrically (properly discontinuously and cocompactly by isometries). Then G is finitely generated and for a fixed x ∈ X the orbit map g 7→ gx is a quasi-isometry. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 15 / 38 The Milnor-Svarc Lemma One of the fundamental tools in studying the coarse geometry of groups is the following fact. Theorem. Let X be a proper metric space and let G act on X geometrically (properly discontinuously and cocompactly by isometries). Then G is finitely generated and for a fixed x ∈ X the orbit map g 7→ gx is a quasi-isometry. The proof involves taking a closed ball K (which is compact, since X is proper) that contains a fundamental domain for the action on X . Let S = {s ∈ G : K ∩ sK 6= ∅}. Since the action is properly discontinuous, this set is finite. We would then show that G = hSi by making a path from x ∈ K to gx ∈ gK for g ∈ G by making steps that are small enough that the corresponding translates of K intersect. These correspond to elements of S. In doing this, the QI bounds fall out. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 15 / 38 The Milnor-Svarc Lemma Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 16 / 38 Consequences of the Milnor-Svarc Lemma If G is a finitely generated group and H ≤ G with [G : H] < ∞, then H y G by multiplication on the right. This action is cocompact, so H is QI to G. This is an example of what is called commensurability (which is stronger than QI). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 17 / 38 Consequences of the Milnor-Svarc Lemma If G is a finitely generated group and H ≤ G with [G : H] < ∞, then H y G by multiplication on the right. This action is cocompact, so H is QI to G. This is an example of what is called commensurability (which is stronger than QI). If G = π1 (M) where M is a compact Riemannian manifold, then G e For instance is finitely generated and is quasi-isometric to M. π1 (S), a closed hyperbolic surface group (χ(S) < 0), is quasi-isometric to the hyperbolic plane. In a similar vein, Zn is quasi-isometric to Rn . Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 17 / 38 QI and Algebra There are many algebraic properties that have geometric content (i.e. are quasi-isometry invariant). In particular: Having a finite presentation is quasi-isometry invariant Having a finite index free subgroup is a quasi-isometry invariant Having two topological ends is equivalent to being virtually Z, so it is a quasi-isometry invariant. Having one end is also a QI invariant. Having a finite index nilpotent subgroup (virtual nilpotence) is equivalent to having polynomial growth (this is Gromov’s Polynomial Growth Theorem). The latter is a quasi-isometry invariant for groups. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 18 / 38 Outline 1 Groups as Geometric Objects 2 Quasi-isometries 3 Hyperbolicity and Hyperbolic Groups Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 19 / 38 Slim Triangle Property Defintion We say that a metric space (X , d) is δ-hyperbolic if for any geodesic triangle [a, b, c] we have that d([a, b], [a, c] ∪ [b, c]) ≤ δ and similarly for all other permutations of the letters. We call such a triangle δ-thin or δ-slim Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 20 / 38 Other Characterizations of Hyperbolicity Hyperbolicity is innately a statement about ”large” triangles in a metric space. In fact, any compact metric space is automatically hyperbolic with δ equal to the (finite) diameter. There are other, equivalent, notions of hyperbolicity. Rips proved that a space is δ-hyperbolic if and only if every quadruple of points a, b, p ∈ X satisfy hb | cip ≥ min{ha | bip , ha | cip } − δ. 1 hx | y ip = (d(x, p) + d(y , p) − d(x, y)) is called the Gromov 2 Product. Bowditch proved in his paper on the curve complex that hyperbolicity can be formulated as a statement about coarse centers of triangles [Bo]. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 21 / 38 Examples of Hyperbolic Metric Spaces As the name suggests, the hyperbolic plane H, which we identify with the complex upper half plan with the Riemannian metric √ dx 2 + dy 2 ds2 = is log(1 + 2)-hyperbolic. Further, hyperbolic 2 y space in any dimension is hyperbolic. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 22 / 38 Examples of Hyperbolic Metric Spaces As the name suggests, the hyperbolic plane H, which we identify with the complex upper half plan with the Riemannian metric √ dx 2 + dy 2 ds2 = is log(1 + 2)-hyperbolic. Further, hyperbolic 2 y space in any dimension is hyperbolic. A tree (of any kind, including an R-tree) is trivially 0-hyperbolic, since any triangle is actually a tripod. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 22 / 38 Quasi-isometric Invariance Proposition. If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also hyperbolic. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 23 / 38 Quasi-isometric Invariance Proposition. If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also hyperbolic. To prove this proposition, we need to study the properties of quasi-isometric images of geodesics. A geodesic is an isometric embedding of an interval. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 23 / 38 Quasi-isometric Invariance Proposition. If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also hyperbolic. To prove this proposition, we need to study the properties of quasi-isometric images of geodesics. A geodesic is an isometric embedding of an interval. A (k, c)-quasigeodesic is a (k , c) quasi-isometric embedding of an interval. Remark. If γ is a geodesic in X and f : X → Y is a (k, c)-quasi-isometric embedding, then f (γ) is a (k, c)-quasigeodesic. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 23 / 38 Quasi-isometric Invariance Definition. A function e : N → R is called a divergence function for a metric (length) space X if for every R, r ∈ N and any pair of geodesics γ : [0, a] → X and γ 0 : [0, a0 ] → X with γ(0) = γ 0 (0) = x, R + r ≤ min{a, a0 } and d(γ(R), γ 0 (R) ≥ e(0), any path connecting γ(R + r ) to γ 0 (R + r ) outside B(x, R + r ) must have length atleast e(r ). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 24 / 38 Quasi-isometric Invariance In the Euclidean plane we must have that e(r ) ≤ πr . Importantly, any divergence function must be linear. As you might think, divergence has to do with the size of a sphere of radius r . Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 25 / 38 Quasi-isometric Invariance In the Euclidean plane we must have that e(r ) ≤ πr . Importantly, any divergence function must be linear. As you might think, divergence has to do with the size of a sphere of radius r . In an infinite tree, the divergence is infinite, since there is only one path between two points. This is called a ”cut point”. H2 has an exponential divergence function, this can be figured out by computing the circumference of a circle of Euclidean radius r centered at 0 in the disc model. Is this also true in other hyperbolic spaces? Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 25 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Proof. Fix R, r ∈ N. Let γ and γ 0 be two geodesics based at some point x ∈ X with d(γ(R), γ(R 0 )) > 2δ and set e(0) = 2δ. Let p be a path in X \ B(x, R + r ) from γ(R + r ) to γ 0 (R + r ). and let α∅ be the geodesic from γ(R + r ) to γ 0 (R + r ). Now let m∅ be the middle point on the path p and let α0 be the geodesic from γ(R + r ) to m∅ and α1 the geodesic from m∅ to γ 0 (R + r ). Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Proof. Now, for any binary string b, let mb be the midpoint of the segment of p between the endpoints of αb . Now let αb0 be the geodesic between the beginning of αb and mb and αb1 the geodesic between mb and the end of αb . Keep subdividing p in this way until each segment in the division has length between 12 and 1. If n is the number of pieces, then log 2 l(p) ≤ n ≤ log 2 l(p) + 1. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Proof. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Proof. For each b, the segments αb , αb0 , αb1 form a geodesic triangle, and so are δ-slim. Since d(γ(R), γ 0 (R)) > δ, there exists a point v (0) on α∅ with d(v (0), γ(R)) < δ. Continuing inductively, we can find v (1) on α0 ∪ α1 with d(v (0), v (1)) ≤ δ. And so if v (i) is on αb we find v (i + 1) on either αb0 or αb1 with d(v (i), v (i + 1)) < δ. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Theorem. If X is a δ-hyperbolic metric space, then it has an exponential divergence function. Proof. Let v (m) be the point obtained at the last level of iteration. There is apoint y ∈ P whose distance from v (m) is at most 1 and so its distance from x is at most R + δ log2 (l(p)) + 2. But d(x, P) ≥ R + r so R + r ≤ R + δ log2 (l(p)) + 2, i.e. l(p) is atleast exponential in r . Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 26 / 38 Quasi-isometric Invariance Proposition. Let γ be a (k , c)-quasigeodesic with end points x and y and let [x, y] denote a geodesic (not necessarily unique) connecting x to y . Then, there exists M = M(k , c, δ) so that the Hausdorff distance between γ and [x, y] is less than M. In particular, γ is in the M-neighborhood of the geodesic between its endpoints The proof of this is an application of the theorem from the previous slide. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 27 / 38 Quasi-isometric Invariance Corollary. If X is a δ-hyperbolic space and f : Y → X be a (K , C)-quasi-isometry, then Y is K (2M + δ) + C-hyperbolic. Proof. Let [a, b, c] be some geodesic triangle in Y and consider its image f ([a, b, c]) in X . Well f ([a, b]) is a quasi-geodesic, so it is in the M-neighborhood of a geodesic [f (a), f (b)] and similarly for [b, c] and [a, c]. Take a point x ∈ f ([a, b]). Let y ∈ [f (a), f (b)] be such that d(x, y ) < M and let z ∈ [f (b), f (c)] ∪ [f (a), f (c)] be such that d(y, z) < δ. Furthermore, there exists w ∈ f ([b, c]) ∪ f ([a, c]) such that d(z, w) < M. So, d(x, w) < 2M + δ. Since f was a quasi-isometry, and x and w are in the image of f , their preimages on the triangle [a, b, c] are at most K (2M + δ) + C apart. Thus the triangle in Y is slim. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 28 / 38 Quasi-isometric Invariance Thanks to Victor Reyes for this image. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 29 / 38 Hyperbolic Groups The previous slide shows us that it makes sense to talk about finitely generated hyperbolic groups. If we transition between finite generating sets, the Cayley graphs are quasi-isometric, so if one is hyperbolic, so is any other. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 30 / 38 Hyperbolic Groups The previous slide shows us that it makes sense to talk about finitely generated hyperbolic groups. If we transition between finite generating sets, the Cayley graphs are quasi-isometric, so if one is hyperbolic, so is any other. e is isometric to Hn ), If M is a closed hyperbolic manifold (i.e. M then π1 (M) is δ-hyperbolic. (This is a consequence of the Milnor-Svarc Lemma) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 30 / 38 Hyperbolic Groups The previous slide shows us that it makes sense to talk about finitely generated hyperbolic groups. If we transition between finite generating sets, the Cayley graphs are quasi-isometric, so if one is hyperbolic, so is any other. e is isometric to Hn ), If M is a closed hyperbolic manifold (i.e. M then π1 (M) is δ-hyperbolic. (This is a consequence of the Milnor-Svarc Lemma) Given a hyperbolic group G, a subgroup H is quasiconvex if and only if the inclusion map H ,→ G is a quasi-isometric embedding. In a hyperbolic group, if g ∈ G is infinite order, then the centralizer of g is quasiconvex. No hyperbolic group contains as Z2 subgroup. In fact, it can not contain a Baumslag-Solitar subgroup a, b | b−1 am b = an . Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 30 / 38 Hyperbolic Groups e is isometric to Hn ), If M is a closed hyperbolic manifold (i.e. M then π1 (M) is δ-hyperbolic. (This is a consequence of the Milnor-Svarc Lemma) Given a hyperbolic group G, a subgroup H is quasiconvex if and only if the inclusion map H ,→ G is a quasi-isometric embedding. In a hyperbolic group, if g ∈ G is infinite order, then the centralizer of g is quasiconvex. No hyperbolic group contains as Z2 subgroup. In fact, it can not contain a Baumslag-Solitar subgroup a, b | b−1 am b = an . [Gromov] If G satisfies the small cancellation condition C 0 ( 61 ), then G is hyperbolic. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 30 / 38 Dehn Presentations It’s natural to wonder if the word problem is solvable in a hyperbolic group (or any new class of groups that is defined). It turns out that hyperbolic groups have very special presentations that make the word problem easy. Definition. Let G = hS | Ri. We say that the presentation is a Dehn presentation if for any reduced word w with w = 1 in G, there exists a relator r ∈ R so that r = r1 r2 , l(r1 ) > l(r2 ) and w = w1 r1 w2 . In other words, any word that represents the identity in G contains more than one half of a relator, and so it can be shortened. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 31 / 38 Dehn Presentations It’s natural to wonder if the word problem is solvable in a hyperbolic group (or any new class of groups that is defined). It turns out that hyperbolic groups have very special presentations that make the word problem easy. Definition. Let G = hS | Ri. We say that the presentation is a Dehn presentation if for any reduced word w with w = 1 in G, there exists a relator r ∈ R so that r = r1 r2 , l(r1 ) > l(r2 ) and w = w1 r1 w2 . In other words, any word that represents the identity in G contains more than one half of a relator, and so it can be shortened. A word which cannot be further reduced or shortened by this method (replacing r1 by r2−1 , a shorter word) is called Dehn reduced. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 31 / 38 Efficient Solution to the Word Problem If G has a finite Dehn presentation (G is finitely generated, R is finite) then you can check all subwords of length at most N = max {l(r ) : r ∈ R} to see if a reduction can be made. This procedure for solving the word problem is called Dehn’s Algorithm, originally created by Max Dehn in 1910 to solve the word problem in surface groups. Its run time is O(|w|2 ) in its simplest iteration. (There are atmost |w| − N subwords in each step, and at most |w| steps in the reduction.) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 32 / 38 Efficient Solution to the Word Problem If G has a finite Dehn presentation (G is finitely generated, R is finite) then you can check all subwords of length at most N = max {l(r ) : r ∈ R} to see if a reduction can be made. This procedure for solving the word problem is called Dehn’s Algorithm, originally created by Max Dehn in 1910 to solve the word problem in surface groups. Its run time is O(|w|2 ) in its simplest iteration. (There are atmost |w| − N subwords in each step, and at most |w| steps in the reduction.) Example. G = ha, b, c, d | [a, b][c, d]i. Let R be the symmetrized set of generators, so that R contains all cyclic conjugates of [a, b][c, d] and its inverse. What about other hyperbolic groups? Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 32 / 38 Hyperbolic Groups: Local Geodesics Definition. A path γ in a metric space X is called a k-local geodesic if every subpath of length k is a geodesic. On a sphere of radius 1, a great circle is a π-local geodesic. If M is a Riemannian manifold, let r (M) be the injectivity radius of M. Then any image of a ray in Tx M under the exponential map is an r (M)-local geodesic. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 33 / 38 Hyperbolic Groups: Local Geodesics Definition. A path γ in a metric space X is called a k-local geodesic if every subpath of length k is a geodesic. On a sphere of radius 1, a great circle is a π-local geodesic. If M is a Riemannian manifold, let r (M) be the injectivity radius of M. Then any image of a ray in Tx M under the exponential map is an r (M)-local geodesic. Local geodesics have to do with loops in a metric space, or relations in a Cayley graph. That makes them natural to study when considering presentations. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 33 / 38 Hyperbolic Groups: Local Geodesics Lemma. Let G be δ-hyperbolic group and let γ be a 4δ-local geodesic. Let g be the geodesic between the endpoints of γ (called γ+ and γ− ). Assume l(g) > 2δ and let r and s be points on γ and g respectively, both distance 2δ from γ+ . Then d(r , s) ≤ δ. The proof here is by induction on the length of γ and uses the thin triangle property multiple times. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 34 / 38 Hyperbolic Groups: Dehn Presentation Theorem. If γ is a 4δ-local geodesic in a δ-hyperbolic group G, then γ is contained in the 3δ neighborhood of the geodesic between its endpoints. We will use this theorem to create a shortening algorithm in our hyperbolic group G. Fix a finite generating set S. Let R = {w : w = 1 in G, |w| < 8δ}. We aim to show that hS | Ri is a Dehn presentation for G. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 35 / 38 Hyperbolic Groups: Dehn Presentation With R as in the previous slide, take a word w in the generators so that w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local geodesic, then it is in the 3δ neighborhood of the origin, so it is already an element of R (since if it had length more than 8δ, the first 4δ long segment of the geodesic would leave the ball of radius 3δ) and we can see that it represents the identity. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 36 / 38 Hyperbolic Groups: Dehn Presentation With R as in the previous slide, take a word w in the generators so that w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local geodesic, then it is in the 3δ neighborhood of the origin, so it is already an element of R (since if it had length more than 8δ, the first 4δ long segment of the geodesic would leave the ball of radius 3δ) and we can see that it represents the identity. Now say that w is not a 4δ-local geodesic. Then there exists some subpath (subword) w1 of length 4δ which is not a geodesic between its endpoints. Replace it in w by the geodesic between its endpoints, call it w2 . Then the path w1 w2−1 has length less than 8δ, so that word is in R. Further, we note that w contained w1 , the longer part of the relator. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 36 / 38 Hyperbolic Groups: Dehn Presentation With R as in the previous slide, take a word w in the generators so that w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local geodesic, then it is in the 3δ neighborhood of the origin, so it is already an element of R (since if it had length more than 8δ, the first 4δ long segment of the geodesic would leave the ball of radius 3δ) and we can see that it represents the identity. Now say that w is not a 4δ-local geodesic. Then there exists some subpath (subword) w1 of length 4δ which is not a geodesic between its endpoints. Replace it in w by the geodesic between its endpoints, call it w2 . Then the path w1 w2−1 has length less than 8δ, so that word is in R. Further, we note that w contained w1 , the longer part of the relator. Now, we can continue this process until we reduce w to a 4δ-local geodesic, a case we have already covered. Thus, G = hS | Ri is a Dehn presentation. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 36 / 38 Open Questions [Be] [Gromov] Given a hyperbolic group G with one topological end (i.e. a freely indecomposable hyperbolic group), does it contain a surface subgroup? Are hyperbolic groups residually finite? [Bestvina] Say that G admits a finite dimensional K (G, 1) and does not contain any Baumslag-Solitar groups. Is G necessarily hyperbolic? If G embeds in a hyperbolic group is this true? (Note: Gromov proved that every hyperbolic group admits a finite dimensional K(G, 1), making this question more natural than it seems.) [Canary] Let H ≤ G, G a hyperbolic group. If there exists some n so that g n ∈ H for every g ∈ G, is H necessarily finite index in G? (The answer is yes is H is quasiconvex) Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 37 / 38 References [Be] Mladen Bestvina, Questions in Geometric Group Theory, http://www.math.utah.edu/ bestvina/eprints/questions-updated.pdf (2004). [Bo] Brian Bowditch, Intersetction Numbers and Hyperbolicity of the Curve Complex, J. reine angew. Math. 598 (2008), 105-129. [BrH] Martin Bridson and Andre Häfliger, Metric Spaces of Non-positive Curvature: Grundlehren der mathematischen Wissenschaften Series, Springer (2010). [Gr] M. Gromov, Hyperbolic Groups: Essays in Group Theory, S. M. Gersten ed., M.S.R.I. Publ 8, Springer (1988), 75 - 263. Tim Susse CUNY Graduate Center () The Coarse Geometry of Groups December 2, 2011 38 / 38