34. Combinatorial Dehn Filling

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 $M$ is 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

$P_i/N_i\to G/\mathcal{N}$ is injective for all $i$ , and
$G/\mathcal{N}$ 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 $i$ .