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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 $M$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

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 $i$.

Today we will see some methods of constructing groups.

Definition. Let $A, B, C$ be groups and let $f:C\to A$ and $g:C\to B$ be injective homomorphisms.  If the diagram below is a pushout then we say write $G=A\mathop{*}_C B$  and we say that $G$ is  the amalgamated (free) product of $A$ and $B$ over $C$.

$\xymatrix{C\ar[r]^f\ar[d]_g&A\ar[d]\\B\ar[r]&G}$

Example. $F_2\cong \mathbb{Z}\mathop{*}_1\mathbb{Z}.$  If $C=1$, we write $G\cong A*B$ and say $G$ is the free product of $A$ and $B$.

As usual, we need to prove existence.

Recall. If $G$ is a group, then the Eilenberg-MacLane Space $K(G,1)$ satisfies the following properties:

• $K(G,1)$ is connected;
• $\pi_1(K(G,1))\cong G$;
• $\pi_1(K(G,1))\cong 1$ for $i\geq2$.

Facts.

1. $K(G,1)$ exists;
2. The construction of $K(G,1)$ is functorial;
3. $K(G,1)$ is unique, up to homotopy equivalence.

For $A,B,C,f,g$ as above, let $X=K(A,1),Y=K(B,1),Z=K(C,1)$ and realize $f$ as a map $\partial_+:Z\to X$ and $g$ as a map $\partial_-:Z\to Y$. Now, let $W=X\sqcup(Z\times[-1,1])\sqcup Y/\sim$, where $(z,\pm1)\sim\partial_\pm(z)$.  By the Seifert-Van Kampen theorem, $\pi_1(W)\cong A\mathop{*}_C B$. Suppose that $A\cong\langle S_1|R_1\rangle$, and $B=\langle S_2|R_2\rangle$.  Then,

$A\mathop{*}_C B\cong\langle S_1\sqcup S_2|R_1,R_2,\{f(c)=g(c)|c\in C\}\rangle$.

In particular, if $A,B$ is finitely generated, then so is $A\mathop{*}_C B$, and if $A,B$ are finitely presented and $C$ is finitely generated, then $A\mathop{*}_C B$ is finitely presented.

Example. Let $\Sigma$ be a connected surface and let $\gamma$ be a separating, simple closed curve.  Let $\Sigma\smallsetminus\mathrm{im}\gamma=\Sigma_+\sqcup\Sigma_+$.  Then,

$\pi_1(\Sigma)\cong\pi_1(\Sigma_-)\mathop{*}_{\langle\gamma\rangle}\pi_1(\Sigma_+).$

But, what if $\gamma$ is non-separating (but still 2-sided)?  Then, there are two natural maps $\partial_\pm:\mathbb{S}^1\to\Sigma_0$ representing $\gamma$, where $\Sigma_0=\Sigma\smallsetminus\mathrm{im}\gamma$.  Associated to $\gamma$, we have a map $i:\pi_1(\Sigma)\to\mathbb{Z}$, $\alpha\mapsto(\alpha\cdot\beta)$, which maps a curve to its signed (algebraic) intersection number with $\gamma$.

Let $\hat\Sigma\to\Sigma$ be a covering map corresponding to $\ker(i)$.  Then,

$\begin{equation*}\pi_1(\hat\Sigma)\cong\cdots\mathop{*}_{\langle\gamma\rangle}\pi_1(\Sigma_0)\mathop{*}_{\langle\gamma\rangle}\pi_1(\Sigma_0)\mathop{*}_{\ ^{\nwarrow}_{\partial_{-*}}\ \langle\gamma\rangle \ _{\partial_{+*}}^{\nearrow}} \pi_1(\Sigma_0)\mathop{*}_{\langle\gamma\rangle} \cdots.\end{equation*}$

This has a shift-automorphism $\tau$.  We can now recover $\pi_1(\Sigma)$:

$\begin{equation*}\pi_1(\Sigma)\cong\pi_1(\hat\Sigma)\mathop{\rtimes}_{\tau}\mathbb{Z}.\end{equation*}$

Defintion. If $f,g:C\to A$ are injective homomorphisms, then let

$\begin{equation*}\hat A=\cdots\mathop{*}_{C} A\mathop{*}_{C} A \mathop{*}_{\ ^{\nwarrow}_{f}\ C \ _{g}^{\nearrow}} A \mathop{*}_{C} \cdots.\end{equation*}$

Let $\tau$ be the shift automorphism on $\hat A$.  Now, $A\mathop*_C=\hat A\mathop\rtimes_\tau\mathbb{Z}$ is called the HNN (Higman, Neumann, Neumann) Extension of $A$ over $C$. We often realize $A\mathop{*}_C$ as $\pi_1(U)$, where $U=X\sqcup(Z\times[-1,1])/\sim$ and $(z,\pm1)\sim\partial_\pm(z)$.  It is easy to write down a presentation:$A\mathop{*}_C\cong\langle S_1,t|R_1,\{tf(c)t^{-1}=g(c)|c\in C\}\rangle$$t$ is called a stable letter.