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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?

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.

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 $l(\Pi_{}(x)) \leq m$.

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

Proof:

Let $y\in[\Pi_{}(x), \Pi_{}(x)]\cap B(1,R)$.  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 $X_2= \Pi_{}^{-1}(\lbrace v^n\mid l(v^n) > M\rbrace)$, where $M$ is provided by Lemma 15.  For all $x_1\in X_1$ we have $l(\Pi_{}(x_1))\leq M$ 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,

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

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 $m,n >>0$.

Therefore, by the Ping-Pong Lemma $\langle u^m, v^n \rangle \cong \mathbb{F}_2$.