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.

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