The Coarse Geometry of Groupstsusse2/talks/Group Theory Talk.pdfTim Susse CUNY Graduate Center The...

The Coarse Geometry of Groups

Tim SusseCUNY 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


The Cayley Graph

Given a presentation of a group G = 〈S | R〉 we associate a geometricobject called the Cayley graph, denoted Cayley(G,S).

Vertex Set = Gg1 ∼ g2 if and only if there exists s ∈ S ∪ S−1 with g1 = g2s

Thus the Cayley graph of a group depends on the choice of generatingset.


The Cayley Graph





The Cayley Graph





Cayley Graph of Z2

Figure: Cayley(Z2, {(1,0), (0,1)})


Cayley Graphs of Z/7Z

(a) (b)

Figure: (a) Cayley(Z/7Z, {1}) (b) Cayley(Z/7Z, {2,3})


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 thegraphic, the two Cayley graphs look very similar. We want to formalizeand take advantage of this.


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 thegraphic, the two Cayley graphs look very similar. We want to formalizeand take advantage of this.


Word Metric on a Group

Given a generating set S for G, we define the distance between twopoints 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 iscalled the word metric on G.

FactsFor any g ∈ G, dS(g,e) = lS(g), the word length of G.For any g,h ∈ G, dS(g,h) = lS(g−1h).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 aproper metric space. The converse is also true.










FactsFor any g ∈ G, dS(g,e) = lS(g), the word length of G.For any g,h ∈ G, dS(g,h) = lS(g−1h).

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 aproper metric space. The converse is also true.





FactsFor any g ∈ G, dS(g,e) = lS(g), the word length of G.For any g,h ∈ G, dS(g,h) = lS(g−1h).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 aproper metric space. The converse is also true.







Outline


2 Quasi-isometries



Bilipschitz Equivalence

Given two metric spaces (X ,dX ) and (Y ,dY ) we say that a mapf : X → Y is a k -bilipschitz map if for any pair of points x1, x2 ∈ X wehave:

1k

dX (x1, x2) ≤ dY (f (x1), f (x2)) ≤ kdX (x1, x2).

A bilipschitz map is like a stretching of the metric at every point (bybounded amounts). It is worth noting the a bilipschitz map is always atopological embedding.

A weaker notion of equivalence is a ”large scale” or coarse bilipschitzcondition. We call this a quasi-isometry.




1k







1k





Quasi-isometric embeddings

Definition. We say that a map f : X → Y is a (k , c) quasi-isometricembedding if for every pair x1, x2 we have that

1k

dX (x1, x2)− c ≤ dY (f (x1), f (x2)) ≤ kdX (x1, x2) + c.

Further, if the map is coarsely onto (i.e. if every point in Y is distanceat most c from f (X )), we call it a quasi-isometry.



Definition. We say that a map f : X → Y is a (k , c) quasi-isometricembedding if for every pair x1, x2 we have that

1k

dX (x1, x2)− c ≤ dY (f (x1), f (x2)) ≤ kdX (x1, x2) + c.

Further, if the map is coarsely onto (i.e. if every point in Y is distanceat most c from f (X )), we call it a quasi-isometry.



Quasi-isometry defines an equivalence relation on metric spaces:Clearly the identity map is an isometry, so any space isquasi-isometric to itself.A quick computation shows that if f and g are quasi-isometries, sois f ◦ g

Symmetry is a little tricky.To show symmetry we need to construct a ”coarse inverse” of aquasi-isometry f . By this we mean a quasi-isometry f−1 so that thereexists a constant r with dX (x , f−1 ◦ f (x)) ≤ r .



Quasi-isometry defines an equivalence relation on metric spaces:Clearly the identity map is an isometry, so any space isquasi-isometric to itself.A quick computation shows that if f and g are quasi-isometries, sois f ◦ gSymmetry is a little tricky.

To show symmetry we need to construct a ”coarse inverse” of aquasi-isometry f . By this we mean a quasi-isometry f−1 so that thereexists a constant r with dX (x , f−1 ◦ f (x)) ≤ r .


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 isbounded. In particular, if f (x1) = f (x2), then dX (x1, x2) ≤ kc. To seethis, look at the left-side of the definition of quasi-isometry:

1k

dX (x1, x2)− c ≤ dY (f (x1), f (x2)) = 0.

So, up to a bounded diameter ”error”, for each y ∈ f (X ), we canchoose 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 ′ ∈ f (X )with dY (y , y ′) ≤ c. Choose some y ′ with this property and let g(y) = xwhere 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.




1k

dX (x1, x2)− c ≤ dY (f (x1), f (x2)) = 0.So, up to a bounded diameter ”error”, for each y ∈ f (X ), we canchoose g(y) = x , where we choose some element x ∈ f−1(y).






1k

dX (x1, x2)− c ≤ dY (f (x1), f (x2)) = 0.So, up to a bounded diameter ”error”, for each y ∈ f (X ), we canchoose g(y) = x , where we choose some element x ∈ f−1(y).




Why is Quasi-isometry Right?

Given a finitely generated group G, there exist many finite generatingsets.

Proposition.Let S and T be two finite generating sets of a group G, then the identitymap 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 Gthere exists a shortest (geodesic) spelling of each ti in the language ofS. In particular, ti = sε1

1i · · · sεkki , where εj = ±1. Let

K = max{k : lS(ti) = k}, i.e. the longest word length of a ti inCayley(G,S). Similarly, let L be the maximum word length of the si inCayley(G,T ).



Given a finitely generated group G, there exist many finite generatingsets.

Proposition.Let S and T be two finite generating sets of a group G, then the identitymap 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 Gthere exists a shortest (geodesic) spelling of each ti in the language ofS. In particular, ti = sε1

1i · · · sεkki , where εj = ±1. Let

K = max{k : lS(ti) = k}, i.e. the longest word length of a ti inCayley(G,S). Similarly, let L be the maximum word length of the si inCayley(G,T ).



Take g ∈ G. Then say lS(g) = r . so g = s1(g) · · · sr (g), where eachsi(g) ∈ S ∪S−1. Replace each of the si with their spellings in t , and wesee that lT (g) ≤ Kr .Similarly, if lT (g) = r , then lS(g) ≤ Lr . So, for any g ∈ G we get:

1K· lT (g) ≤ lS(g) ≤ L · lT (g).

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-Lipschitzequivalence.

Notice in our Cayley graphs for Z before we had K = 3, L = 2 forS = {1} and T = {2,3}.



Take g ∈ G. Then say lS(g) = r . so g = s1(g) · · · sr (g), where eachsi(g) ∈ S ∪S−1. Replace each of the si with their spellings in t , and wesee that lT (g) ≤ Kr .Similarly, if lT (g) = r , then lS(g) ≤ Lr . So, for any g ∈ G we get:

1K· lT (g) ≤ lS(g) ≤ L · lT (g).

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-Lipschitzequivalence.

Notice in our Cayley graphs for Z before we had K = 3, L = 2 forS = {1} and T = {2,3}.


The Milnor-Svarc Lemma

One of the fundamental tools in studying the coarse geometry ofgroups 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 isfinitely generated and for a fixed x ∈ X the orbit map g 7→ gx is aquasi-isometry.

The proof involves taking a closed ball K (which is compact, since X isproper) that contains a fundamental domain for the action on X . LetS = {s ∈ G : K ∩ sK 6= ∅}. Since the action is properly discontinuous,this set is finite.

We would then show that G = 〈S〉 by making a path from x ∈ K togx ∈ gK for g ∈ G by making steps that are small enough that thecorresponding translates of K intersect. These correspond to elementsof S. In doing this, the QI bounds fall out.



One of the fundamental tools in studying the coarse geometry ofgroups 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 isfinitely generated and for a fixed x ∈ X the orbit map g 7→ gx is aquasi-isometry.

The proof involves taking a closed ball K (which is compact, since X isproper) that contains a fundamental domain for the action on X . LetS = {s ∈ G : K ∩ sK 6= ∅}. Since the action is properly discontinuous,this set is finite.

We would then show that G = 〈S〉 by making a path from x ∈ K togx ∈ gK for g ∈ G by making steps that are small enough that thecorresponding translates of K intersect. These correspond to elementsof S. In doing this, the QI bounds fall out.




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 iscocompact, so H is QI to G. This is an example of what is calledcommensurability (which is stronger than QI).

If G = π1(M) where M is a compact Riemannian manifold, then Gis finitely generated and is quasi-isometric to M. For instanceπ1(S), a closed hyperbolic surface group (χ(S) < 0), isquasi-isometric to the hyperbolic plane.In a similar vein, Zn is quasi-isometric to Rn.


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 iscocompact, so H is QI to G. This is an example of what is calledcommensurability (which is stronger than QI).If G = π1(M) where M is a compact Riemannian manifold, then Gis finitely generated and is quasi-isometric to M. For instanceπ1(S), a closed hyperbolic surface group (χ(S) < 0), isquasi-isometric to the hyperbolic plane.In a similar vein, Zn is quasi-isometric to Rn.


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 invariantHaving a finite index free subgroup is a quasi-isometry invariantHaving two topological ends is equivalent to being virtually Z, so itis a quasi-isometry invariant. Having one end is also a QIinvariant.Having a finite index nilpotent subgroup (virtual nilpotence) isequivalent to having polynomial growth (this is Gromov’sPolynomial Growth Theorem). The latter is a quasi-isometryinvariant for groups.


Outline


2 Quasi-isometries



Slim Triangle Property

Defintion We say that a metric space (X ,d) is δ-hyperbolic if for anygeodesic triangle [a,b, c] we have that d([a,b], [a, c] ∪ [b, c]) ≤ δ andsimilarly for all other permutations of the letters. We call such a triangleδ-thin or δ-slim


Other Characterizations of Hyperbolicity

Hyperbolicity is innately a statement about ”large” triangles in a metricspace. In fact, any compact metric space is automatically hyperbolicwith 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 everyquadruple of points a,b,p ∈ X satisfy

〈b | c〉p ≥ min{〈a | b〉p , 〈a | c〉p} − δ.

〈x | y〉p =12

(d(x ,p) + d(y ,p)− d(x , y)) is called the GromovProduct.Bowditch proved in his paper on the curve complex thathyperbolicity can be formulated as a statement about coarsecenters of triangles [Bo].


Examples of Hyperbolic Metric Spaces

As the name suggests, the hyperbolic plane H, which we identifywith the complex upper half plan with the Riemannian metric

ds2 =dx2 + dy2

y2 is log(1 +√

2)-hyperbolic. Further, hyperbolic

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.


Examples of Hyperbolic Metric Spaces

As the name suggests, the hyperbolic plane H, which we identifywith the complex upper half plan with the Riemannian metric

ds2 =dx2 + dy2

y2 is log(1 +√

2)-hyperbolic. Further, hyperbolic

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.


Quasi-isometric Invariance

Proposition.If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is alsohyperbolic.

To prove this proposition, we need to study the properties ofquasi-isometric images of geodesics.

A geodesic is an isometric embedding of an interval.A (k , c)-quasigeodesic is a (k , c) quasi-isometric embedding ofan 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.





A geodesic is an isometric embedding of an interval.

A (k , c)-quasigeodesic is a (k , c) quasi-isometric embedding ofan 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.





A geodesic is an isometric embedding of an interval.A (k , c)-quasigeodesic is a (k , c) quasi-isometric embedding ofan 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.



Definition. A function e : N→ R is called a divergence function for ametric (length) space X if for every R, r ∈ N and any pair of geodesicsγ : [0,a]→ X and γ′ : [0,a′]→ X with γ(0) = γ′(0) = x ,R + r ≤ min{a,a′} and d(γ(R), γ′(R) ≥ e(0), any path connectingγ(R + r) to γ′(R + r) outside B(x ,R + r) must have length atleast e(r).



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 onepath between two points. This is called a ”cut point”.H2 has an exponential divergence function, this can be figured outby computing the circumference of a circle of Euclidean radius rcentered at 0 in the disc model.Is this also true in other hyperbolic spaces?



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 onepath between two points. This is called a ”cut point”.H2 has an exponential divergence function, this can be figured outby computing the circumference of a circle of Euclidean radius rcentered at 0 in the disc model.Is this also true in other hyperbolic spaces?



Theorem.If X is a δ-hyperbolic metric space, then it has an exponentialdivergence function.

Proof.




Proof.Fix R, r ∈ N. Let γ and γ′ be two geodesics based at some point x ∈ Xwith d(γ(R), γ(R′)) > 2δ and set e(0) = 2δ.Let p be a path in X \ B(x ,R + r) from γ(R + r) to γ′(R + r). and let α∅be the geodesic from γ(R + r) to γ′(R + r). Now let m∅ be the middlepoint on the path p and let α0 be the geodesic from γ(R + r) to m∅ andα1 the geodesic from m∅ to γ′(R + r).




Proof.Now, for any binary string b, let mb be the midpoint of the segment of pbetween the endpoints of αb. Now let αb0 be the geodesic between thebeginning of αb and mb and αb1 the geodesic between mb and the endof αb. Keep subdividing p in this way until each segment in the divisionhas length between 1

2 and 1. If n is the number of pieces, thenlog 2l(p) ≤ n ≤ log 2l(p) + 1.




Proof.




Proof.For each b, the segments αb, αb0, αb1 form a geodesic triangle, and soare δ-slim.Since d(γ(R), γ′(R)) > δ, there exists a point v(0) on α∅ withd(v(0), γ(R)) < δ.Continuing inductively, we can find v(1) on α0 ∪ α1 withd(v(0), v(1)) ≤ δ. And so if v(i) is on αb we find v(i + 1) on either αb0or αb1 with d(v(i), v(i + 1)) < δ.




Proof.Let v(m) be the point obtained at the last level of iteration. There isapoint y ∈ P whose distance from v(m) is at most 1 and so itsdistance 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 .



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 γand [x , y ] is less than M.In particular, γ is in the M-neighborhood of the geodesic between itsendpoints

The proof of this is an application of the theorem from the previousslide.



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 imagef ([a,b, c]) in X . Well f ([a,b]) is a quasi-geodesic, so it is in theM-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 thatd(x , y) < M and let z ∈ [f (b), f (c)] ∪ [f (a), f (c)] be such thatd(y , z) < δ. Furthermore, there exists w ∈ f ([b, c]) ∪ f ([a, c]) such thatd(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 , theirpreimages on the triangle [a,b, c] are at most K (2M + δ) + C apart.Thus the triangle in Y is slim.



Thanks to Victor Reyes for this image.


Hyperbolic Groups

The previous slide shows us that it makes sense to talk about finitelygenerated hyperbolic groups. If we transition between finitegenerating sets, the Cayley graphs are quasi-isometric, so if one ishyperbolic, so is any other.

If M is a closed hyperbolic manifold (i.e. M is isometric to Hn),then π1(M) is δ-hyperbolic. (This is a consequence of theMilnor-Svarc Lemma)Given a hyperbolic group G, a subgroup H is quasiconvex if andonly if the inclusion map H ↪→ G is a quasi-isometric embedding.In a hyperbolic group, if g ∈ G is infinite order, then the centralizerof g is quasiconvex.No hyperbolic group contains as Z2 subgroup. In fact, it can notcontain a Baumslag-Solitar subgroup

⟨a,b | b−1amb = an⟩.

[Gromov] If G satisfies the small cancellation condition C′(16), then

G is hyperbolic.


Hyperbolic Groups


If M is a closed hyperbolic manifold (i.e. M is isometric to Hn),then π1(M) is δ-hyperbolic. (This is a consequence of theMilnor-Svarc Lemma)

Given a hyperbolic group G, a subgroup H is quasiconvex if andonly if the inclusion map H ↪→ G is a quasi-isometric embedding.In a hyperbolic group, if g ∈ G is infinite order, then the centralizerof g is quasiconvex.No hyperbolic group contains as Z2 subgroup. In fact, it can notcontain a Baumslag-Solitar subgroup



G is hyperbolic.


Hyperbolic Groups





G is hyperbolic.


Hyperbolic Groups




G is hyperbolic.


Dehn Presentations

It’s natural to wonder if the word problem is solvable in a hyperbolicgroup (or any new class of groups that is defined). It turns out thathyperbolic groups have very special presentations that make the wordproblem easy.Definition. Let G = 〈S | R〉. We say that the presentation is a Dehnpresentation if for any reduced word w with w = 1 in G, there exists arelator r ∈ R so that r = r1r2, l(r1) > l(r2) and w = w1r1w2.

In other words, any word that represents the identity in G containsmore 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 r−1

2 , a shorter word) is called Dehn reduced.


Dehn Presentations

It’s natural to wonder if the word problem is solvable in a hyperbolicgroup (or any new class of groups that is defined). It turns out thathyperbolic groups have very special presentations that make the wordproblem easy.Definition. Let G = 〈S | R〉. We say that the presentation is a Dehnpresentation if for any reduced word w with w = 1 in G, there exists arelator r ∈ R so that r = r1r2, l(r1) > l(r2) and w = w1r1w2.

In other words, any word that represents the identity in G containsmore 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 r−1

2 , a shorter word) is called Dehn reduced.


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 mostN = max {l(r) : r ∈ R} to see if a reduction can be made.

This procedure for solving the word problem is called Dehn’sAlgorithm, originally created by Max Dehn in 1910 to solve the wordproblem in surface groups. Its run time is O(|w |2) in its simplestiteration. (There are atmost |w | − N subwords in each step, and atmost |w | steps in the reduction.)

Example. G = 〈a,b, c,d | [a,b][c,d ]〉. Let R be the symmetrized set ofgenerators, so that R contains all cyclic conjugates of [a,b][c,d ] andits inverse. What about other hyperbolic groups?


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 mostN = max {l(r) : r ∈ R} to see if a reduction can be made.

This procedure for solving the word problem is called Dehn’sAlgorithm, originally created by Max Dehn in 1910 to solve the wordproblem in surface groups. Its run time is O(|w |2) in its simplestiteration. (There are atmost |w | − N subwords in each step, and atmost |w | steps in the reduction.)

Example. G = 〈a,b, c,d | [a,b][c,d ]〉. Let R be the symmetrized set ofgenerators, so that R contains all cyclic conjugates of [a,b][c,d ] andits inverse. What about other hyperbolic groups?


Hyperbolic Groups: Local Geodesics

Definition. A path γ in a metric space X is called a k -local geodesic ifevery 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 ofM. Then any image of a ray in TxM under the exponential map isan r(M)-local geodesic.

Local geodesics have to do with loops in a metric space, or relations ina Cayley graph. That makes them natural to study when consideringpresentations.



Definition. A path γ in a metric space X is called a k -local geodesic ifevery 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 ofM. Then any image of a ray in TxM under the exponential map isan r(M)-local geodesic.

Local geodesics have to do with loops in a metric space, or relations ina Cayley graph. That makes them natural to study when consideringpresentations.



Lemma.Let G be δ-hyperbolic group and let γ be a 4δ-local geodesic. Let g bethe geodesic between the endpoints of γ (called γ+ and γ−). Assumel(g) > 2δ and let r and s be points on γ and g respectively, bothdistance 2δ from γ+. Then d(r , s) ≤ δ.

The proof here is by induction on the length of γ and uses the thintriangle property multiple times.


Hyperbolic Groups: Dehn Presentation

Theorem.If γ is a 4δ-local geodesic in a δ-hyperbolic group G, then γ iscontained in the 3δ neighborhood of the geodesic between itsendpoints.

We will use this theorem to create a shortening algorithm in ourhyperbolic group G. Fix a finite generating set S. LetR = {w : w = 1 in G, |w | < 8δ}. We aim to show that 〈S | R〉 is a Dehnpresentation for G.



With R as in the previous slide, take a word w in the generators so thatw = 1 in G. If the loop w in the Cayley graph is already a 4δ-localgeodesic, then it is in the 3δ neighborhood of the origin, so it is alreadyan element of R (since if it had length more than 8δ, the first 4δ longsegment of the geodesic would leave the ball of radius 3δ) and we cansee that it represents the identity.

Now say that w is not a 4δ-local geodesic. Then there exists somesubpath (subword) w1 of length 4δ which is not a geodesic between itsendpoints. Replace it in w by the geodesic between its endpoints, callit w2. Then the path w1w−1

2 has length less than 8δ, so that word is inR. 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-localgeodesic, a case we have already covered. Thus, G = 〈S | R〉 is aDehn presentation.














Open Questions [Be]

[Gromov] Given a hyperbolic group G with one topological end(i.e. a freely indecomposable hyperbolic group), does it contain asurface subgroup?Are hyperbolic groups residually finite?[Bestvina] Say that G admits a finite dimensional K (G,1) anddoes not contain any Baumslag-Solitar groups. Is G necessarilyhyperbolic? If G embeds in a hyperbolic group is this true? (Note:Gromov proved that every hyperbolic group admits a finitedimensional K(G, 1), making this question more natural than itseems.)[Canary] Let H ≤ G, G a hyperbolic group. If there exists some nso that gn ∈ H for every g ∈ G, is H necessarily finite index in G?(The answer is yes is H is quasiconvex)


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 theCurve Complex, J. reine angew. Math. 598 (2008), 105-129.

[BrH] Martin Bridson and Andre Hafliger, Metric Spaces ofNon-positive Curvature: Grundlehren der mathematischenWissenschaften 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.


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