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18. Graphs of groups and graphs of spaces

8 March 2009 in Course notes | Tags: Graphs of groups | by pweil | 3 comments

We still need to convince ourselves of some basic facts about the previous lecture, for example is the map $A \to A *_C B$ injective?

Example: Cut the sphere $S^2$ along the equator. Then the diagram we have is

$\xymatrix{\mathbb{Z}\ar[r] \ar[d] &1 \ar[d]\\1\ar[r]&1}$

Definition: If $G=A *_C B$ or $G = A*_C$ we say that G splits over C, and we call C the edge group. If $G=A*_C$ or $G = A *_C B$ and $C$ is not $A$ or $B$ in the latter case, then we say that $G$ splits non-trivially.

Definition: Let $\Gamma$ be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex $v \in V(\Gamma)$ (resp. edge $e \in E(\Gamma)$ ) let $G_v$ (resp. $G_e$ ) be a group. If $v_\pm$ are vertices adjoining an edge e then let $\partial_\pm^e : G_e \to G_{v_\pm}$ be an injective homomorphism. This data determines a graph of groups $\mathcal G$ .

We say that $\mathcal G$ has:

underlying graph $\Gamma$
vertex groups $\{ G_v \}$
edge groups $\{ G_e \}$
edge maps $\{ \partial_\pm^e \}$

Similarly, we have:

Definition: Let $\Xi$ be a connected graph. For each vertex $v \in V(\Xi)$ (resp. $e \in E(\Xi)$ ) let $X_v$ (resp. $X_e$ ) be a connected CW-complex. If $v_\pm$ adjoin $e$ let $\partial_\pm^e : X_e \to X_{v_\pm}$ be $\pi_1$ -injective continuous maps. This data determines a graph of spaces $\mathcal X$ . It has underlying graph $\Xi$ , vertex spaces $X_v$ , edge spaces $X_e$ , etc. The graph of spaces $\mathcal X$ determines a space as follows: define

$\displaystyle X_{\mathcal X} = \left(\bigsqcup_{v \in V(\Xi)} X_v \sqcup \bigsqcup_{e\in E(\Xi)} (X_e \times [-1,+1])\right) / \sim,$

where $(x,\pm 1) \sim \partial_\pm^e(x)$ for $x \in X_e$ . We say that $\mathcal{X}$ is a graph-of-spaces structure (or decomposition) for $X_\mathcal{X}$ .

Remark: There is a natural map $X_\mathcal{X} \to \Xi$ (by collapsing all the edge and vertex spaces).

Given any graph of groups $\mathcal G$ we can construct a graph of spaces $\mathcal X$ with underlying graph $\Gamma$ by assigning $X_v = K(G_v,1), X_e = K(G_e,1)$ and realizing the edge maps as continuous maps $X_e \to X_{v_\pm}$ . We write $X_\mathcal G$ for $X_\mathcal X$ . This is well-defined up to homotopy equivalence.

Definition: The fundamental group of $\mathcal G$ is just $\pi_1(\mathcal G) = \pi_1(X_\mathcal{G}).$

Examples:

If $\mathcal{G} = \xymatrix{A{\bullet} \ar@{-}[r]^C &{\bullet} B}$ then $\pi_1 (G) = A *_C B$
If then $\pi_1 (G) = A *_C$
Let $\mathcal C \subset \Sigma$ be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along $\mathcal C$ decomposes $\Sigma$ into a graph of spaces and $\pi_1 (\Sigma)$ into a graph of groups.

Note: The edge maps of $X_\mathcal{G}$ are only defined up to free (i.e. unbased) homotopy. Translated to $\mathcal G$ , this means that only the conjugacy class of $\partial_\pm$ in $G_{v_\pm}$ matters.

Remark: The map $X_\mathcal G \to \Gamma$ induces a surjection $\pi_1(\mathcal G) \to \pi_1(\Gamma).$

Here’s a way to construct a graph of groups. Let’s suppose $G$ acts on a tree $T$ without edge inversions (we can do this by subdividing edges if necessary). Let $Y = \widetilde{K(G,1)}$ . The group $G$ acts diagonally on $T \times Y.$ The quotient $X = G\setminus (T \times Y)$ has a structure of a graph of spaces. The underlying graph is $\Gamma = G\setminus T$ and there is a natural map $X \to \Gamma$ .

Let $v \in V(\Gamma)$ be a vertex below $\tilde v \in V(T)$ . The preimage of $v$ is just $v \times (G_{\tilde v} \setminus Y)$ where $G_{\tilde v}$ is the stabilizer of $\tilde v$ . Similarly, for $e \in E(\Gamma)$ below $\tilde e \in E(T)$ , the preimage of $e$ is $e \times (G_{\tilde e} \setminus Y).$

If $\tilde e$ adjoins $\tilde v$ then $G_{\tilde e} \subset G_{\tilde v}$ so the edge map $G_{\tilde e} \setminus Y \to G_{\tilde v} \setminus Y$ is a covering map and therefore $\pi_1$ -injective. We have defined a graph of spaces $\mathcal X$ and $\pi_1(X_\mathcal X) = G$ since $T \times Y$ is simply connected.

Applying $\pi_1$ to everything, we have a graph of groups $\mathcal G$ . Its underlying graph is $\Gamma$ . Its vertex groups are the vertex stabilizers of $T$ , its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, $\pi_1(\mathcal G) = G$ .

Question for next time: Does every graph of groups arise in this way?

17. New Groups From Old

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

16. Free Subgroups of Hyperbolic Groups (from Sam Kim)

3 March 2009 in Course notes | Tags: Free groups, Hyperbolic groups | by elandes | 2 comments

Theorem 12 (Gromov): Let $\Gamma$ be torsion-free $\delta$ -hyperbolic group. If $u,v \in\Gamma$ such that $uv\neq vu$ , then for all sufficiently large $m,n$ , $\langle u^m,v^n\rangle \cong F_2$ .

Remark: The torsion-free hypothesis is not necessary, but it allows us to avoid some technicalities. For instance, it is a non-obvious fact that an infinite hyperbolic group contains a copy of $\mathbb{Z}$ .

For the rest of this lecture $\Gamma$ will be a torsion-free $\delta$ -hyperbolic group, $uv\neq vu$ where $u,v$ are primitive (i.e. not proper powers).

Recall that for $\Gamma$ torsion-free $\delta$ -hyperbolic, $u$ primitive implies that $\langle u \rangle = C(u)= C(u^m)$ .

If $u$ and $v$ do not commute we can show there is some point $u^p$ on $\langle u \rangle$ arbitrarily far from $\langle v \rangle$ .
Hence we have the following lemma.

Lemma 13: $d_{haus}(\langle u \rangle , \langle v \rangle )=\infty$

If $u$ and $v$ do not commute there is some point $u^p$ on $\langle u\rangle$ arbitrarily far from $\langle v\rangle$ .

Proof: Suppose not. That means $\exists R_0 > 0$ such that $\forall u^p \in \langle u \rangle \exists v^q \in \langle v \rangle$ such that $d(u^p,v^q) = d(1,u^{-p}v^q) < R_0$ . So $u^{-p}v^q$ is in $B(1,R_0)$ . But the Cayley graph is locally finite so $B(1,R_0)$ has finitely many elements. By the Pigeonhole Principle $\exists p\neq r$ such that $u^{-p}v^q=u^{-r}v^s$ for some $q, s$ . Then $\langle u \rangle = C(u) =C(u^{p-r})=C(v^{q-s})=C(v)=\langle v \rangle$ . But then $uv=vu$ . $\Rightarrow\Leftarrow$ .

For a moment view $\langle u \rangle$ and $\langle v \rangle$ as the horizontal and vertical geodesics in $\mathbb{H}$ . For two points $x$ on $\langle u \rangle$ and $y$ on $\langle v \rangle$ , we can argue that the geodesic between them curves toward the origin.

And so we have Lemma 14.

Lemma 14: There exists $R > 0$ such that $\forall m,n$ , $[u^m,v^n]\cap B(1,R)\neq \emptyset$ .

Proof:

Recall that $\phi : \mathbb{Z}\to \Gamma$ by $\phi (i)= u^i$ is a quasi-isometric embedding. So by Theorem 6, $d_{haus}({1,u,u^2,\dots, u^m},[1,u^m]) < R_1$ and $d_{haus}({1,v,v^2,\dots, v^n},[1,v^n]) < R_1$

By Lemma 13 choose $u^p \in \langle u \rangle$ such that
$d(u^p,\langle v \rangle) > 2R_1 + \delta$ . Choose $u_p \in [1,u^m]$ such that $d(u_p,u^p) < R_1$ . Now, $u_p$ must be $\delta$ -close to $[u^m,v^n]$ so for some point $x$ on the geodesic between $v^n$ and $u^m$ , $d(u_p, x) < \delta$ . Then $d(1,[u^m,v^n])\leq d(1,u^p) + d(u^p,u_p) + d(u_p, x) \leq l(u^p)+R_1 + \delta$ . $\Box$

For a subgroup $H \subseteq \Gamma$ , one can choose a closest point projection $\Pi_H : \Gamma \to H$ which is $H$ -equivariant. (Write $\Gamma = \cup H{g_i}$ . Choose $\Pi_H(g_i)=h_i$ where $h_i$ and $g_i$ are close and declare $\Pi_H$ to be $H$ -equivariant.) $\Pi_{H}$ is typically not a group homomorphism.

We’re interested in $\Pi_{\langle u\rangle}$ and $\Pi_{\langle v\rangle}$ .
In $\mathbb{H}^2$ , there is some $m$ such that $\forall x\in \mathbb{H}^2$ either $l(\Pi_{}(x)) \leq m$ or $l(\Pi_{<v>}(x)) \leq m$ .

Lemma 15: $\exists M$ such that $\forall x\in Cay(\Gamma)$ , $l(\Pi_{}(x)) \leq M$ or $l(\Pi_{<v>}(x)) \leq M$ .

Proof:

Let $y\in[\Pi_{}(x), \Pi_{<v>}(x)]\cap B(1,R)$ . WLOG, $y$ is $\delta$ -close to $p\in[x,\Pi_{}(x)]$ and $d(1, \Pi_{}(x) \leq d(1,p)+d(p,\Pi_{}(x)) \leq d(1,p) +d(p,1)$ since $\Pi_{}(x)$ is the closest point to $x$ (in particular compared to $u^0=1$ ). So $d(1,\Pi_{}(x)) \leq 2d(1,p)\leq 2(R+\delta)$ . $\Box$ .

Now we can prove the theorem.

Proof of Theorem 12:

The idea is to use the Ping-Pong Lemma on the Cayley graph.

Let $X_1 = \Pi_{}^{-1}(\lbrace u^m\mid l(u^m) > M\rbrace)$ and let $X_2= \Pi_{<v>}^{-1}(\lbrace v^n\mid l(v^n) > M\rbrace)$ , where $M$ is provided by Lemma 15. For all $x_1\in X_1$ we have $l(\Pi_{<v>}(x_1))\leq M$ and likewise for all $x_2\in X_2$ we have $l(\Pi_{}(x_2))\leq M$ . In particular, $X_1 \cap X_2 = \emptyset$ .

Let $x_2\in X_2$ . By $\langle u\rangle$ -equivariance,

$\Pi_{}(u^m x_2)=u^m\Pi_{} (x_2)$

for any $m$ . In particular,

$l(\Pi_{}(u^m x_2))\geq l(u^m)-l(\Pi_{}(x_2))\geq l(u^m)-M$

by the triangle inequality. Similarly,

$l(\Pi_{<v>}(v^n x_1))\geq l(v^n)-l(\Pi_{<v>}(x_1))\geq l(v^n)-M$

for all $x_1\in X_1$ and all $n$ . Because $\langle u\rangle$ and $\langle v\rangle$ are quasi-isometrically embedded, it follows that $u^mX_2 \subset X_1$ and $v^n X_1\subset X_2$ for $m,n >>0$ .