You are currently browsing pweil’s articles.

Last time: Theorem 21 (Groves–Manning–Osin): If $G$ is hyperbolic rel $\mathcal P$ then there exists a finite subset $A \subseteq G\setminus 1$ such that if $\bigcup_i N_i \cap A = \emptyset$ then
(a) $P_i/N_i \to G/\mathcal N$ is injective;
(b) $G$ is hyperbolic rel $P_i/N_i$.

Theorem 22 (Gromov, Olshanshkii, Delzant): If $G$ is hyperbolic relative to the infinite cyclic $\{\langle g_1\rangle,\dots,\langle g_n \rangle\}$ then there is a $K>0$ such that for all $l_1,\dots,l_n>0$ there exists a $\phi : G \to G'$ hyperbolic such that $o(\phi(g_i))=Kl_i$ for each $i$.

The proof is an easy application of Groves–Manning–Osin.

Definition: If $\{\langle g_1 \rangle,\dots,\langle g_n\rangle\}$ (infinite cyclic) is malnormal then we say $g_1,\dots,g_n$ are independent. A group G is omnipotent if for every independent $g_1,\dots,g_n$ there exists a $K>0$ such that for all $l_1,\dots,l_n>0$ there exists a homomorphism $\phi$ from $G$ to a finite group such that $o(\phi(g_i)) = Kl_i$ for all $i$.

Omnipotence strengthens residual finiteness for torsionfree groups.

Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.

We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.

Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup $H$ of any hyperbolic group $G$ is separable.

Let $g \in G \setminus H$. The idea is to Dehn fill $H$ to get a new hyperbolic group $\bar G$ in which the image $\bar H$ is finite and $\bar g \not\in\bar H$. If we could do this, we would be done by residual finiteness. This works if $H$ is malnormal. But it probably isn’t. Fortunately, we can quantify how far $H$ is from being malnormal:

Definition: The height of $H$ is the maximal $n \in \mathbb N$ such that there are distinct cosets $g_1 H,\dots,g_n H \in G/H$ such that the intersection
$g_1 H g_1^{-1} \cap \dots \cap g_n H g_n^{-1}$
is infinite.

H is height $0$ iff $H$ is finite. In a torsionfree group, $H$ is height $1$ iff $H$ is malnormal.

Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.

Agol, Groves and Manning are able to prove:

Theorem 24: Let $G$ be a (torsionfree) residually finite hyperbolic group, and $H$ a quasiconvex subgroup of height $k$. Let $g \in G\setminus H$. Then is an epimorphism $\eta: G \to \bar G$ to a hyperbolic group such that
(i) $\eta(H)$ is quasiconvex in $\bar G$;
(ii) $\eta(g) \not\in\eta(H)$;
(iii) $\eta(H)$ has height $\leq k-1$.

The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of $H$. Theorem 22 is an easy consequence.

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?