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34. Combinatorial Dehn Filling

20 April 2009 in Course notes | Tags: Hyperbolic groups, Relatively hyperbolic groups | by jmaciejewski | Leave a comment

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 $i$ .

17. New Groups From Old

5 March 2009 in Course notes | Tags: Graphs of groups | by jmaciejewski | 7 comments

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 $C$ .

$http://www.codecogs.com/eq.latex?\xymatrix{C\ar[r]^f\ar[d]_g&A\ar[d]\\B\ar[r]&G}$

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.

$K(G,1)$ exists;
The construction of $K(G,1)$ is functorial;
$K(G,1)$ 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,

$http://www.codecogs.com/eq.latex?$A\mathop{*}_C%20B\cong\langle%20S_1\sqcup%20S_2|R_1,R_2,\{f(c)=g(c)|c\in%20C\}\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,

$http://www.codecogs.com/eq.latex?$%20\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,

$http://www.codecogs.com/eq.latex?\begin{equation*}\pi_1(\hat\Sigma)\cong\cdots\mathop{*}_{\langle\gamma\rangle}\pi_1(\Sigma_0)\mathop{*}_{\langle\gamma\rangle}\pi_1(\Sigma_0)\mathop{*}_{\%20^{\nwarrow}_{\partial_{-*}}\%20\langle\gamma\rangle%20%20\%20_{\partial_{+*}}^{\nearrow}}%20\pi_1(\Sigma_0)\mathop{*}_{\langle\gamma\rangle}%20\cdots.\end{equation*}$

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

$http://www.codecogs.com/eq.latex?\begin{equation*}\pi_1(\Sigma)\cong\pi_1(\hat\Sigma)\mathop{\rtimes}_{\tau}\mathbb{Z}.\end{equation*}$

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

$http://www.codecogs.com/eq.latex?\begin{equation*}\hat%20A=\cdots\mathop{*}_{C}%20A\mathop{*}_{C}%20A%20\mathop{*}_{\%20^{\nwarrow}_{f}\%20C%20%20\%20_{g}^{\nearrow}}%20A%20\mathop{*}_{C}%20%20\cdots.\end{equation*}$

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.