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Recall, that for any graph \Gamma we built a combinatorial horoball \mathcal{H}(\Gamma).  For a group G and a collection of subgroups \mathcal{P}=\{P_1,\ldots,P_n\}and a generating set S, we built the augmented Cayley graph X by gluing copies of \mathcal{H}(\mathrm{Cay}(G)).  G is hyperbolic relative to \mathcal{P} if and only if X is Gromov hyperbolic.

Exercise 28: If A and B are finitely generated, then A*B is hyperbolic relative \{A,B\}. (Hint: X is a graph of spaces with underlying graph a tree and the combinatorial horoballs for vertex spaces.)

Example: Suppose Mis a complete hyperbolic manifold of finite volume.  So, \Gamma=\pi_1M acts on \mathbb{H}^n.  Let \Lambda be a subset of \partial\mathbb{H}^n consisting of points that are the unique fixed point of some element of \Gamma.  So \Gamma acts on \Lambda, and there only finitely many orbits.  Let P_1,\ldots,P_n be stabilizers of representatives from these orbits and let \mathcal{P}=\{P_1,\ldots,P_n\}.  Then, \Gamma is hyperbolic relative to \mathcal{P}.

Example: Let G be a torsion-free word-hyperbolic group.  Then, G is clearly hyperbolic relative to \{1\}.  A collection of subgroups P_1,ldots,P_n is malnormal if for any g\in G, P_i\cap gP_jg^{-1}\neq1 implies that i=j and g\in P_i.  G is hyperbolic relative to \mathcal{P}=\{P_1,\ldots,P_n\} if and only if \mathcal{P} is malnormal.

The collection of subgroups \mathcal{P} is the collection of peripheral subgroups.

Lemma 31: If G is torsion-free and hyperbolic relative to a set of quasiconvex subgroups \mathcal{P}, then \mathcal{P} is malnormal.

Sketch of Proof: Suppose that P_1\cap gP_2g^{-1} is infinite.  Consider the following rectangles:  Note that if k=l(g), then gP_2g^{-1} is contained in a k-neighborhood of gP_2.  Now, there exists infinite sequences p_i\in P_1 and q_i\in P_2 such that d(p_i,gq_i)\leq k.  Look at the rectangles with vertices 1, g, gp_i, p_i.  The geodesics in X between 1 and p_i and g and gq_i go arbitrarily deep into the combinatorial horoballs.  Therefore, they are arbitrarily far apart.  It follows that these rectangles cannot be uniformly slim.

Let \mathcal{N}=\{N_1,\ldots,N_n\} where each N_i\lhd P_i.  Write G/\langle\langle\bigcup_iN_i\rangle\rangle=G/\mathcal{N}.  Call this the Dehn filling of G.

Note: If G is hyperbolic relative to \mathcal{P}, then G is hyperbolic.

Theorem 21: (Groves-Manning-Osin). Suppose G is hyperbolic relative to \mathcal{P}.  Then, there exists a finite set A contained in G\smallsetminus 1 such that whenever (\bigcup_i N_i)\cap A\neq\emptyset we have

  1. P_i/N_i\to G/\mathcal{N} is injective for all i, and
  2. G/\mathcal{N} is hyperbolic relative to the collection \{P_i/N_i\};

In particular, if P_i/N_i are all hyperbolic, then so is G/\mathcal{N}.

One application of this theorem is a simple proof of a theorem of Gromov, Olshanskii, and Delzant:

Theorem 22: Let G be hyperbolic and suppose \{\langle g_1\rangle,\ldots,\langle g_n\rangle\} is malnormal, with each \langle g_i\rangle infinite.  Then, there is constant K such that for all positive integers l_1,\ldots,l_n there is an epimorphism to a hyperbolic group \phi:G\to G' such that o(\phi(g_i))=Kl_i for each i.

Some intuition: Recall that if M is a closed hyperbolic manifold
then \pi_1(M) is word-hyperbolic. However, a lot of interesting hyperbolic manifolds are not closed.

Example: Let K\subset S^3 be the figure 8 knot.

figure 1

Then the complement M_{8}=S^{3} K admits a complete hyperbolic metric and is of finite volume.

So, here we have an example of a hyperbolic manifold which is not compact but is of finite volume. This is almost as which is almost as natural as being closed.

M_{8} is homotopy equivalent to M_{8}', the complement of a thickened K in S^{3}.

fig2

M_8' is a compact manifold with boundary and its interior admits a hyperbolic metric. The boundary of M_8' is homeomorphic to a 2-torus, so \partial M_8' \hookrightarrow M_8' induces a map \mathbb{Z}^2\hookrightarrow\pi_1M_8' . By Dehn’s lemma, the map is injective so \pi_1M_8' cannot be word hyperbolic. The point is that \pi_1M_8 acts nicely on \mathbb{H}^2 but no cocompactly so the Svarc=Milnor lemma does not apply.

The torus boundary component of M_8' corresponds to a cusp of M_8.

The point is that we can use cusped manifolds like M_8' to build a lot of manifolds and in particular a lot of hyperbolic manifolds.

Take M_8' and a solid Torus T .

Choose a homeomorphism \phi: \partial M_8' \hookrightarrow\partial T

Definition: The manifold M_{\phi}=M_{8}'\cup_{\phi}T is obtained from M_{8}' by Dehn filling .

We now want to understand what we have done to \pi_{1}M_{8}. The map \phi induces a map \phi_{*}:

lecture_4_17_09_xymatrix1

The surjectivity of \phi_{*} follows from the fact that \phi is a homeomorphism. The Seifert Van Kampen theorem implies that \pi_{1}M_{\phi}=\pi_{1}M_{8}\langle\langle \ker(\phi_{*})\rangle\rangle, where \langle\langle\ker(\phi_{*}) \rangle\rangle denotes the normal closure of \ker(\phi_{*}) .

Gromov-Thurston 2\pi theorem: Let M be any compact hyperbolic manifold and \partial_{0}M be a component of \partial M homeomorphic to a 2-torus for all but finitely many choices of

lecture_4_17_09_xymatrix2

the Dehn filling M_{\phi} is hyperbolic.

Note: by finitely many we mean finitely many maps up to homotopy.

This is a very fruitful way of building hyperbolic manifolds. The next question to ask is whether we can do the same thing for groups. So, now we will try to develop a group theoretic version of this picture.

Let \Gamma be a group theoretic graph with the induced length metric. Construct a new graph \mathcal{H}(\Gamma) called the combinatorial horoball on \Gamma as follows: Define the vertices V(\mathcal{H})=V(\Gamma)\times \mathbb{N}. There are two sorts of edges in {E}(\mathcal{H}). We say that (u,k) and (v,k) are joined by a (horizontal) edge if d_{\Gamma}(u,v) \leq 2^{k} and u\neq v. We say that (v,k) and (v,k+1) are joined by a (vertical) edge for all k.

fig6
For k large enough u' and v' will have distance one and L\leq 1 iff 2^{k} \ge d_{\Gamma}(u,v) iff k\leq \log_{2}d_{\Gamma}(u,v).

Exercise 27:
(A). For u,vin V(\Gamma), d_{\mathcal{H}}((u,0),(v,0))\approx \log_{2}d_{\Gamma}(u,v).

(B). For any connected \Gamma, \mathcal{H}(\Gamma) is Gromov hyperbolic .

fig7

Let G be a group and let \mathcal{P}=\{ P_{1},\ldots, P_{n} \} be a finite set of finitely generated subgroups of G. Choose a finite generating set S for G such that for each i, s_i=S \cap P_i generate P_i. Then \mathrm{Cay}(G,S) contains natural copies of \mathrm{Cay}(P_{i},S_{i}).

Construct the augmented Cayley graph X=X(G,\mathcal{P},S) by gluing on combinatorial horoballs equivariantly.

X(G,\mathcal{P},S) = \mathrm{Cay}(G,S) \cup \bigcup_{i} \lbrack \mathcal{H}(\mathrm{Cay}(P_{i},S_{i})) \times G/P_{i} \rbrack / \sim where for each i
and each gP_{i}\in G /P_{i}, \mathcal{H}(\mathrm{Cay}(P_{i},S_{i}) \times \{ gP_{i}\} is glued to g\mathrm{Cay}(P_{i},S_{i}) along \mathrm{Cay}(P_{i},S_ {i}) \times \{ 0 \} .

Definition: G is hyperbolic rel \mathcal{P} if and only if X(G,P,S) is Gromov hyperbolic for some (any) choice of S.

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