392C Geometric Group Theory

35. Separating quasi-convex subgroups.

22 April 2009 in Uncategorized | by pweil | Leave a comment

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.

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 $i$ .

33. Relatively Hyperbolic Groups

19 April 2009 in Course notes | Tags: Relatively hyperbolic groups | by elandes | Leave a comment

Some intuition: Recall that if $M$ is a closed hyperbolic manifold
then $\pi_1(M)$ is word-hyperbolic. However, a lot of interesting hyperbolic manifolds are not closed.

Example: Let $K\subset S^3$ be the figure 8 knot.

Then the complement $M_{8}=S^{3}$ $K$ admits a complete hyperbolic metric and is of finite volume.

So, here we have an example of a hyperbolic manifold which is not compact but is of finite volume. This is almost as which is almost as natural as being closed.

$M_{8}$ is homotopy equivalent to $M_{8}'$ , the complement of a thickened $K$ in $S^{3}$ .

fig2

$M_8'$ is a compact manifold with boundary and its interior admits a hyperbolic metric. The boundary of $M_8'$ is homeomorphic to a 2-torus, so $\partial M_8' \hookrightarrow M_8'$ induces a map $\mathbb{Z}^2\hookrightarrow\pi_1M_8'$ . By Dehn’s lemma, the map is injective so $\pi_1M_8'$ cannot be word hyperbolic. The point is that $\pi_1M_8$ acts nicely on $\mathbb{H}^2$ but no cocompactly so the Svarc=Milnor lemma does not apply.

The torus boundary component of $M_8'$ corresponds to a cusp of $M_8$ .

fig3

The point is that we can use cusped manifolds like $M_8'$ to build a lot of manifolds and in particular a lot of hyperbolic manifolds.

Take $M_8'$ and a solid Torus $T$ .

fig4
Choose a homeomorphism $\phi: \partial M_8' \hookrightarrow\partial T$

Definition: The manifold $M_{\phi}=M_{8}'\cup_{\phi}T$ is obtained from $M_{8}'$ by Dehn filling .

We now want to understand what we have done to $\pi_{1}M_{8}$ . The map $\phi$ induces a map $\phi_{*}$ :

lecture_4_17_09_xymatrix1

The surjectivity of $\phi_{*}$ follows from the fact that $\phi$ is a homeomorphism. The Seifert Van Kampen theorem implies that $\pi_{1}M_{\phi}=\pi_{1}M_{8}\langle\langle \ker(\phi_{*})\rangle\rangle$ , where $\langle\langle\ker(\phi_{*}) \rangle\rangle$ denotes the normal closure of $\ker(\phi_{*})$ .

Gromov-Thurston $2\pi$ theorem: Let M be any compact hyperbolic manifold and $\partial_{0}M$ be a component of $\partial M$ homeomorphic to a 2-torus for all but finitely many choices of

the Dehn filling $M_{\phi}$ is hyperbolic.

Note: by finitely many we mean finitely many maps up to homotopy.

This is a very fruitful way of building hyperbolic manifolds. The next question to ask is whether we can do the same thing for groups. So, now we will try to develop a group theoretic version of this\picture.

fig5

Let $\Gamma$ be a group theoretic graph with the induced length metric. Construct a new graph $\mathcal{H}(\Gamma)$ called the combinatorial horoball on $\Gamma$ as follows: Define the vertices $V(\mathcal{H})=V(\Gamma)\times \mathbb{N}$ . There are two sorts of edges in ${E}(\mathcal{H})$ . We say that $(u,k)$ and $(v,k)$ are joined by a (horizontal) edge if $d_{\Gamma}(u,v) \leq 2^{k}$ and $u\neq v$ . We say that $(v,k)$ and $(v,k+1)$ are joined by a (vertical) edge for all $k$ .

fig6
For $k$ large enough $u'$ and $v'$ will have distance one and $L\leq 1$ iff $2^{k} \ge d_{\Gamma}(u,v)$ iff $k\leq \log_{2}d_{\Gamma}(u,v)$ .

Exercise 27:
(A). For $u,vin V(\Gamma)$ , $d_{\mathcal{H}}((u,0),(v,0))\approx \log_{2}d_{\Gamma}(u,v)$ .

(B). For any connected $\Gamma$ , $\mathcal{H}(\Gamma)$ is Gromov hyperbolic .

fig7

Let $G$ be a group and let $\mathcal{P}=\{ P_{1},\ldots, P_{n} \}$ be a finite set of finitely generated subgroups of $G$ . Choose a finite generating set $S$ for $G$ such that for each $i$ , $s_i=S \cap P_i$ generate $P_i$ . Then $\mathrm{Cay}(G,S)$ contains natural copies of $\mathrm{Cay}(P_{i},S_{i})$ .

Construct the augmented Cayley graph $X=X(G,\mathcal{P},S)$ by gluing on combinatorial horoballs equivariantly.

$X(G,\mathcal{P},S) = \mathrm{Cay}(G,S) \cup \bigcup_{i} \lbrack \mathcal{H}(\mathrm{Cay}(P_{i},S_{i})) \times G/P_{i} \rbrack / \sim$ where for each $i$
and each $gP_{i}\in G$ / $P_{i}$ , $\mathcal{H}(\mathrm{Cay}(P_{i},S_{i}) \times \{ gP_{i}\}$ is glued to $g\mathrm{Cay}(P_{i},S_{i})$ along $\mathrm{Cay}(P_{i},S_ {i}) \times \{ 0 \}$ .

Definition: G is hyperbolic rel $\mathcal{P}$ if and only if $X(G,P,S)$ is Gromov hyperbolic for some (any) choice of $S$ .

32. Grushko’s Theorem

16 April 2009 in Course notes | Tags: Graphs of groups | by kczar44 | Leave a comment

Definition. A group $G$ splits freely if $G$ acts on a tree $T$ without global fixed point and such that every edge stailizer is trivial. If $G$ does not split freely, then $G$ is called freely indecomposable.

Example. $\mathbb Z=\pi_1(S^1)$ . Equivalently, $\mathbb Z$ acts on $\mathbb R$ without global fixed points. So $\mathbb Z$ splits freely.

If $G \ncong \mathbb Z$ but $G$ splits freely, then $G=G_1 \ast G_2$ for $G_1, G_2 neq 1$ .

Definition. The rank of $G$ is the minimal $r$ such that $F_r$ surjects $G$ .

It is clear that $rank(G_1\ast G_2)\leq rank(G_1)+rank(G_2)$ .

Grushko’s Lemma. Suppose $\varphi:F_r \longrightarrow G$ is surjective and $r$ is minimal. If $G=G_1 \ast G_2$ , then $F_r=F_1 \ast F_2$ such that $\varphi(F_i)=G_i$ for $i=1,2$ .

Pf. Let $X_i=K(G_i,1) (i=1,2)$ be simplicial and let $\mathfrak{X}$ be a graph of spaces with vertex spaces $X_1, X_2$ and edge space a point. So $G=\pi_1(X_{\mathfrak{X}}, x_0)$ where $x_0=(*, \frac{1}{2})$ .

Let $\Gamma$ be a graph so that $\pi_1(\Gamma)\cong F_r$ and realize $\varphi$ as a simplicial map $f: \Gamma \longrightarrow X_{\mathfrak{X}}$ . Let $y_0 \in f^{-1}(x_0)$ . Because $r$ is minimal, $f^{-1}(x_0)$ is a forest, contained in $\Gamma$ . The goal is to modify $f$ by a homotopy to reduce the number of connected components of $f^{-1}(x_0)$ .

Let $U \subseteq f^{-1}(x_0)$ be the component that contains $y_0$ . Let $V \subseteq f^{-1}(x_0)$ be some other component. Let $\alpha$ a path in $\Gamma$ from $y_0$ to $V$ .

Look at $f \circ \alpha \in \pi_1(X_{\mathfrak{X}}, x_0)$ . Because $\varphi$ is surjective, there exists $\gamma\in \pi_1(\Gamma, y_0)$ such that $f \circ \gamma = f \circ \alpha$ . Therefore if $\beta= \gamma^{-1} \cdot \alpha$ , then $f \circ \beta$ is null-homotopic in $X_{\mathfrak{X}}$ and $\beta$ gives a path from $y_0$ to $V$ .

We can write $\beta$ as a concaternation as $\beta=\beta_1 \cdot \beta_2 \cdot \cdot \cdot \beta_n$ such that for each $i$ , $f \circ \beta_i \subseteq X_{\mathfrak{X}}\smallsetminus {x_0}$ . By the Normal Form Theorem, there exists $i$ such that $f\circ \beta_i$ is null-homotopic in $X$ .

We can now modify $f$ by a homotopy so that $im (f\circ\beta_i)={x_0}$ . Therefore $\beta_i \subseteq f^{-1}(x_0)$ and the number of components of $f^{-1}(x_0)$ has gone down. By induction, we can choose $f$ so that $f^{-1}(x_0)$ is a tree. Now $f$ factors through $\Gamma'=\Gamma/ f^{-1}(x_0)$ . Then $F_r\cong \pi_1(\Gamma')$ and there is a unique vertex of $\Gamma'$ that maps to $x_0$ . So every simple loop in $\Gamma'$ is either contained in $X_1$ or $X_2$ as required. $square$

An immediate consequence is that $rank(G_1\ast G_2)=rank (G_1) + rank (G_2)$ .

Grushko’s Theorem. Let $G$ be finitely generated. Then $G\cong G_1 \ast \cdots\ast G_m \ast F_r$ where each $G_i$ is freely indecomposable and $F_r$ is free. Furthermore, the integers $m$ and $r$ are unique and the $G_i$ are unique up to conjugation and reordering.

Pf. Existence is an immediate corollary of the fact that rank is additive.

Suppose $G=H_1\ast \cdots \ast H_n \ast F_s$ . Let $\mathcal{G}$ be the graph of groups. Let $T$ be the Bass-Serre tree of $\mathcal{G}$ .

Consider the action of $G_i$ on $T$ . Because $G_i$ is freely indecomposable, $G_i$ stabilize a vertex of $T$ . Therefore $G_i$ is conjugate into some $H_i$ .

Now consider the action of $F_r$ on $T$ . $F_r\smallsetminus T$ is a graph of groups with underlying graph $\Delta$ , say, and $\pi_1(\Delta)$ is a free factor in $F_r$ . But there is a covering map $F_r\smallsetminus T \longrightarrow \mathcal{G}$ that induces a surjection $\pi_1(\Delta) \longrightarrow F_s$ . Therefore, $r\geq s$ . The other inequality can be obtained by switching $F_r$ and $F_s$ . $\square$

31. Doubles & virtual retractions

15 April 2009 in Course notes | Tags: Doubles of free groups, Virtual retractions | by smhart | Leave a comment

Last time, we used the following lemma without justification, so let’s prove it now.

Lemma 30. Let $\mathcal{G}$ be a graph of groups with $\Gamma$ finite and $G = \pi_1 \mathcal{G}$ finitely generated. If $G_e$ is finitely generated for every edge $e \in E ( \Gamma )$ , then $G_v$ is finitely generated for every $v \in V ( \Gamma )$ .

This is not completely trivial: it is certainly possible for finitely generated group to have subgroups that are not finitely generated. Indeed, recall the proof that every countable group embeds in the free group on three generators.

Pf. Let $S$ be a finite generating set for $G$ , and for each $e \in E ( \Gamma )$ let $S_e$ be a finite generating set for the edge group $G_e$ . By the Normal Form Theorem, every $g \in S$ can be written in the form

$g = g_0 t_1^{\pm 1} g_1 \cdots t_n^{\pm 1} g_n$

where each $t_i$ is a stable letter and each $g_i \in G_{v_i}$ for some $v_i \in V ( \Gamma )$ . For a fixed $v \in V ( \Gamma )$ , let

$\displaystyle S_v = \bigcup_{g\in S} \{ g_i : g_i \in G_v \} \cup \bigcup_{e \text{ adjoining } v} \partial_e^{\pm} ( S_e )$ ,

where for each $e \in E ( \Gamma )$ adjoining $v$ the plus or minus is chosen so that $\partial_e^\pm : G_e to G_v$ . It is clear then that $S_v$ is contained in $G_v$ . To see that $S_v$ is finite, note that since $S$ is finite, the first union is finite; and since $\Gamma$ is finite there can be only finitely many edges adjoining a given vertex, so the second union is finite. Hence it remains to prove that $S_v$ generates $G_v$ .

Let $\gamma \in G_v$ . Because $S$ generates $G$ , we have

$\gamma = \gamma_1 \cdots \gamma_m$

where $\gamma_j \in S$ . Each $\gamma_j$ has a normal form as above, so we get an expression of the form

$\gamma = g_0 t_1^{\pm 1} g_1 t_n^{\pm 1} g_n$ , or $1 = \gamma^{-1} g_0 t_1^{\pm 1} g_1 t_n^{\pm 1} g_n$ .

By the Normal Form Theorem, the expression on the right can then be simplified. After the simplification process, we have no stable letters left, and every $g_i$ is either contained in $G_v$ or is a product of elements of the incident edge groups, and in both cases lie in $S_v$ . $\square$

Remember that Theorem 19 said $D$ is LERF, answering our question (b). But in fact, we get more from the proof of Theorem 19.

Definition. Recall that $H \subset G$ is a retract if the inclusion $H \hookrightarrow G$ has a left inverse $\rho : G \to H$ . Similarly, we call $H$ a virtual retract if $H$ is a retract of a finite index subgroup of $G$ .

For instance, Marshall Hall’s Theorem implies that every finitely generated subgroup of a free group is a virtual retract.

Theorem 20. Every finitely generated subgroup of $D$ is a virtual retract.

Pf. Consider the setup of the proof of Theorem 19. We start with a subgroup $H = \pi_1 X_{\mathfrak{X}'}$ and end up with a finitely sheeted covering space $X_{\widehat{\mathfrak{X}}} \to X_{\mathfrak{X}'}$ . The graph of spaces $\widehat{\mathfrak{X}}$ is built using the “obvious” bijection between elevations to $\mathfrak{X}^+$ and elevations to $\mathfrak{X}^-$ . Thus the identification $X_{\mathfrak{X}^+} \to X_{\mathfrak{X}^-}$ extends to a topological retraction $X_{\widehat{\mathfrak{X}}} \to X_{\mathfrak{X}^+}$ . Now, we shall build a map $\pi_1 X_{\mathfrak{X}^+} \to \pi_1 X_{\mathfrak{X}'} = H$ . We build it at each vertex space, one at a time. From the proof of Lemma 29, we see that the core of each $X_{v'}$ is a topological retract of the corresponding $X_{v^+}$ . Furthermore, we can choose the retraction so that for each long loop of degree $d$ that we added is mapped to a null-homotopic loop in $X_{v'}$ . This allows you to piece together the map $X_{v^+} \to X_{v'}$ into a retraction $\pi_1 X_{\mathfrak{X}^+} \to \pi_1 X_{\mathfrak{X}'} = H$ . $\square$

Exercise 14 asserted that a (virtual) retract is quasi-isometrically embedded, and so Theorem 20 has the following corollary:

Corollary. Every finitely generated subgroup of $D$ is quasi-convex.

30. Doubles of Free Groups (continued)

13 April 2009 in Course notes | Tags: Doubles of free groups, Elevations, Separability | by dgirao | Leave a comment

Lemma 29: Suppose $\{f_i':C_i\longrightarrow\Gamma^{H}\}$ is a finite set of infinite degree elvations and $\Delta \subseteq \Gamma^{H}$ is compact. Then for all sufficiently large $d>0$ , there exists an intermediate covering $\Gamma_d$ such that

(a) $\Delta$ embeds in $\Gamma_d$

(b) every $f'_i$ descends to an elevation $\hat{f_i}:\hat{C_i}\longrightarrow \Gamma_d$ of degree $d$

(c) the $\hat{f_i}$ are pairwise distinct

Proof: We claim that the images of $f_i'$ never share an infinite ray (a ray is an isometric embedding of $[0,\infty)$ ). Neither do two ends of the same elevation $f_i'$ . Let’s claim by passing to the universal cover of $\Gamma$ , a tree $T$ .

diagram

For each $i$ , lift $f_i'$ to a map $\tilde{f_i}:\mathbb{R}\longrightarrow T$ . If $f'_i$ and $f'_j$ share an infinite ray then there exists $h\in H$ such that $\tilde{f_i}$ and $h\tilde{f_j}$ overlay in an infinite ray. The point is that $\tilde{f_i}, \tilde{f_j}$ correspond to cosets $g_if_{\ast}(\pi_1(C))$ and $g_jf_{\ast}(\pi_1(C))$ . But this implies that

$g_if_{\ast}(\pi_1(C))=hg_jf_{\ast}(\pi_1(C))$

This implies that $Hg_if_{\ast}(\pi_1(C))=Hg_jf_{\ast}(\pi_1(C))$ . So $f'_i=f'_j$ . A similar argument implies that the two ends of $f'_i$ do not overlap in an infinite ray. This proves the claim.

Let $\Gamma'$ be the core of $\Gamma^{H}$ . Enlarging $\Delta$ if necessary, we can assume that

(i) $\Gamma'\subseteq\Delta$ ;

(ii) $\Delta$ is a connected subgraph;

(iii) for each $i$ , for some $x_i\in C'_i$ , $f'_i(x_i)\in\Delta$ ;

(iv) for each $i$ , $|im(f'_i)\cap \delta\Delta|=2$ .

For each $i$ identifying $C'_i$ with $\mathbb{R}$ so that $C$ is identified with $\mathbb{R}/\mathbb{Z}$ and $x_i$ is identified with $0$ . Let

$\Delta_d=\Delta\cup(\bigcup_i f'_i([-d/2,d/2]))$

For all sufficiently large $d$ ,

$f'_i(\pm d/2)\notin\Delta$

Now, the restriction of $\Delta_d \longrightarrow \Delta$ factors through $\Delta_d/\sim\longrightarrow\Gamma$ , where $f'_i(d/2)\sim f'_i(-d/2)$ . This is a finite-to-one immersion, so, by theorem 5, we can complete it to a finite-sheeted covering map as required. $\Box$

Theorem 19: $D$ is LERF.

Recall the set-up from the previous lecture. We built a graph of spaces $\mathscr{X}$ for $D$ .

Proof: Let $H\subset D$ be finitely generated. Let $X_H$ be the corresponding covering space of $X_{\mathscr{X}}$ and let $\Delta\subseteq X_H$ be compact. Because $H$ is finitely generated, there exists a subgraph of spaces $X'$ such that $\pi_1(X') =H$ . We can take $X'$ large enough so that $\Delta \subseteq X'$ . We can enlarge $\Delta$ so that it contains every finite-degree edge space of $X'$ . Also enlarge $\Delta$ so that

$\partial^{\pm}_{e'}(\Delta\times (\mathrm{interval}))\subseteq\Delta$

for any $e'\in E(\Xi')$ . For each $v'\in V(\Xi')$ let $\Delta_{v'}=\Delta\cap X_{v'}$ and let ${f'_i}=\{$ incident edge map of infinite degree $\}$ .

Applying lemma 29 to $\Gamma^H=X_{v'}$ , for some large $d$ , set $X_{\hat{v}}=\Gamma_d$ . (Here we use the fact that vertex groups of $\mathscr{X}'$ are finitely generated)

Define $\mathscr{X}^+$ as follows:

$\bullet \Xi^+=\Xi^-$

$\bullet$ For each $v^+\in V(\Xi^+)$ , the edge space is the $X_{v^+}$ that the lemma produced from the corresponding $v'$ .

Now, by construction, $\bigcup_{v^+}X_{v^+}$ can be completed to a graph of spaces $\mathscr{X}^+$ so that the map

$X_{\mathscr{X}'}\longrightarrow X_{\mathscr{X}}$

factors through $X_{\mathscr{X}'}\longrightarrow X_{\mathscr{X}}$ and $\Delta$ embeds. Let $\mathscr{X}^-$ be identical to $\mathscr{X}^+$ except with +’s and -’s exchanged. Clearly $\mathscr{X}^+\cup\mathscr{X}^-$ satisfies Stallings condition, as required. $\Box$

29. Doubles of free groups

8 April 2009 in Course notes | Tags: Doubles of free groups, Elevations, Separability | by yyao392c | Leave a comment

Agol-Groves-Manning’s Theorem predicts that, for every word-hyperbolic group we can easily construct, every quasiconvex subgroup is separable (otherwise, we would find a non-residually finite hyperbolic group!).

In this section, we use graphs of groups to build new hyperbolic groups:

Combination Theorem (Bestvina & Feighn): If $H$ is a malnormal subgroup of hyperbolic groups $G_1, G_2$ , then $G_1\ast_H G_2$ is hyperbolic.

Recall: $H$ is called a malnormal subgroup of $G$ if it satisfies: if $gHg^{-1}\cap H\neq 1$ , then $g\in H$ .

For a proof, see M. Bestvina and M. Feighn, “A combination theorem for negatively curved groups”, J. Differential Geom., 35 (1992), 85–101.

Example: Let $F$ be free, $w\in F$ not a proper power. By Lemma 11, $\langle w\rangle\leq F$ is malnormal, so $D:=F\ast_{\langle w\rangle} F$ is hyperbolic. As a special case, if $\Sigma$ is closed surface of even genus $n=2k$ , considered as the connected sum of two copies of the closed surface of genus $k$ , then by Seifert-van Kampen Theorem, $\pi_1(\Sigma)=F_{2k}\ast_{\langle w\rangle} F_{2k}$ for some $w\in F_{2k}$ .

Question: (a) Which subgroups of $D$ are quasiconvex? (b) Which subgroups of $D$ are separable?

We will start by trying to answer (b). The following is an outline of the argument: Let $\Gamma$ be a finite graph so that $\pi_1(\Gamma)=F$ , let $\Gamma_{\pm}$ be two copies of $\Gamma$ . Realize $w\in F$ as maps $\partial^{\pm}: C\rightarrow \Gamma_{\pm}$ , where $C\simeq S^1$ . Let $X$ be the graph of spaces with vertex spaces $\Gamma_{\pm}$ , edge space $C$ , and edge maps $\partial^{\pm}$ . Then clearly, $D\simeq \pi_1(X)$ , and finitely generated subgroups $H\leq D$ are in correspondence with covering spaces $X^H\rightarrow X$ . We can then use similar technique to sections 27 and 28.

Let us now make a few remarks about elevations of loops. Let $f: C\rightarrow X$ be a loop in some space $X$ , i.e., $C\simeq S^1$ and $\pi_1(C)\simeq\mathbf{Z}$ . Consider an elevation of $f$ :

diagram

The conjugacy classes of subgroups of $\mathbf{Z}$ are naturally in bijection with $\mathbf{N}\cup\{\infty\}$ . The degree of the elevation is equal to the degree of the covering map $C'\rightarrow C$ .

Definition: Suppose $X'\rightarrow X$ is a covering map and $\widehat{X}$ is an intermediate covering space, i.e., $X'\rightarrow X$ factors through $\widehat{X}\rightarrow X$ , and we have a diagram

diagram1

If $f'$ and $\widehat{f}$ are elevations of $f$ and the diagram commutes, then we say that $f'$ descends to $\widehat{f}$ .

Let $\Gamma$ be a finite graph, $H\leq \pi_1(\Gamma)$ a finitely generated subgroup and $f: C\rightarrow\Gamma$ a loop. Let $\Gamma^H\rightarrow\Gamma$ be a covering space corresponding to $H$ .

Lemma 29: Consider a finite collection of elevations $\{f'_i: C'_i\rightarrow\Gamma^H\}$ of $f$ to $\Gamma^H$ , each of infinite degree. Let $\Delta\leq\Gamma^H$ be compact. Then for all sufficiently large $d>0$ , there exists an intermediate, finite-sheeted covering space $\Gamma_d\rightarrow\Gamma$ satisfying: (a) $\Delta$ embeds in $\Gamma_d$ ; (b) every $f'_i$ descends to an elevation $\widehat{f}_i: \widehat{C}_i\rightarrow\Gamma_d$ of degree exactly $d$ ; (c) these $\widehat{f}_i$ are pairwise distinct.

28. Separability properties of amalgams

8 April 2009 in Course notes | Tags: Graphs of groups, Residual finiteness | by drosen2000 | Leave a comment

As before, are $\chi'$ and $\chi$ are graphs of spaces equipped with maps $\Phi \colon X_{\chi'} \to X_{\chi}$ , $\varXi' \to \varXi$ , $\phi_{v'} \colon X_{v'} \to X_v$ , and $\phi_{e'} \colon X_{e'} \to X_e$ such that

fig_28_11

commutes.

Lemma 28: Suppose that every edge map of $\chi'$ is an elevation. Then the map $\Phi$ is $\pi_1$ -injective.

Proof: The idea is to add extra vertex spaces to $\chi'$ so that $\chi'$ satisfies Stalling’s condition. As before, we have inclusions: fig_28_2

If $\chi'$ does not satisfy Stalling’s condition then one of these maps is not surjective. Without loss of generality, suppose that fig_28_31

does not arise as an edge map of $\chi'$ . Suppose $\partial_{e}^{-} \colon X_e \to X_u$ . Then $(\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'}))$ is a subgroup of $\pi_1(X_u)$ . Let $X_{u'}$ be the corresponding covering space of $X_u$ . Since $\pi_1(X_{u'})$ contains $(\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'}))$ , there is a lift $\partial_{e'} \colon X_{e'} \to X_{u'}$ of $\partial_{e'} \circ \phi_{e'}$ . Now replace $\chi'$ by $\chi' \cup X_{u'}$ and repeat. After infinitely many repetitions, the result $\hat{\chi}$ satisfies the hypothesis of Stalling’s condition, and $\chi'$ is contained in a subgraph of spaces.

Theorem 18: If $G_{+}$ and $G_{-}$ are residually finite groups then $G_{+} * G_{-}$ is also residually finite.

Proof: Let $X_{\pm} = K(G_{\pm}, 1)$ , let $e = *$ be a point, and fix maps $\partial^{\pm} \colon e \to X_{\pm}$ . This defines a graph of spaces $\chi$ . By the Seifert-van Kampen Theorem, $\pi_1(X_{\chi}) \cong G_{+} * G_{-}$ . Let $\tilde{\chi}$ be the graph of spaces structure on the universal cover of $X_{\chi}$ , and let $\Delta \subseteq X_{\tilde{\chi}}$ be a compact subset. We may assume that $\Delta$ is connected; we may also assume that if $\tilde{e} \in \tilde{\varXi}$ (Bass-Serre tree) and $\Delta \cap \tilde{e} \ne \emptyset$ , then $\tilde{e} \subseteq \Delta$ . Let $\eta \colon X_{\tilde{\chi}} \to \tilde{\varXi}$ be the map to the underlying map of the Bass-Serre tree. Then $\eta(\Delta)$ is a finite connected subgraph, $\varXi'$ . For each $v' \subseteq v(\varXi')$ , let $\Delta_{v'} = \Delta \cap X_{v'}$ , a compact subspace of $X_{v'}$ . Because $G_{\pm}$ are residually finite, we have a diagram fig_28_6

where $X_{\hat{v}} \to X_{\pm}$ is a finite-sheeted covering map and $\Delta_{v'}$ embeds in $X_{\hat{v}}$ . Let $\hat{\chi}$ be defined as follows. Set $\hat{\varXi} = \varXi'$ ; for a vertex $\hat{v} \in \hat{\varXi}$ , the vertex space is the $X_{\hat{v}}$ corresponding to $v'$ . For each $\hat{e} \in E(\hat{\varXi})$ corresponding to $e' \in E(\varXi')$ , define $\partial_{\hat{e}}^{\pm}$ so that the diagram fig_28_81

commutes. Now sum.

Exercise 26: If $G_1$ and $G_2$ are residually finite and $H$ is finite, prove that $G_1 *_H G_2$ is residually finite.

27. Stallings’ Condition

6 April 2009 in Course notes | Tags: Elevations, Graphs of groups, Separability | by mathjoker | 1 comment

Recall that our goal is to determine, for a given map of graphs of spaces such as the one shown below, whether the map $\Phi$ can be extended to a covering map $\widehat{\Phi}$ .

Figure 1

Let $\mathcal{X}, \mathcal{X}'$ be graphs of spaces equipped with maps $\Xi' \to \Xi$ , $\phi_{v'} : X_{v'} \to X_v$ and $\phi_{e'} : X_{e'} \to X_e$ as before. Recall that in Stallings’ proof of Hall’s Theorem, we completed an immersion to a covering map by gluing in edges. We will aim to do the same thing with graphs of spaces.

Definition: Let $\mathcal{X}$ be a graph of spaces, and let $\eta : X_{\mathcal{X}} \to \Xi$ be the map to the underlying graph. If $\Delta \subseteq \Xi$ is a subgraph, then $\eta^{-1}(\Delta) \subseteq X_{\mathcal{X}}$ has a graph-of-spaces structure $\mathcal{Y}$ with underlying graph $\Delta$ . Call $\mathcal{Y}$ a subgraph of spaces of $\mathcal{X}$ .

We’re seeking a condition on $\mathcal{X}'$ such that $\mathcal{X}'$ is realized as a subgraph of spaces of some $\widehat{\mathcal{X}}$ with a covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ such that the following diagram commutes:

diagram1 Definition: For each edge map $\partial_e^{\pm} : X_e \to X_v$ of $\mathcal{X}$ , and each $v' \mapsto v$ a vertex of $\mathcal{X}'$ , let

$\mathcal{E}^{\pm}(e) = \bigcup_{v' \mapsto v} \mathcal{E}^{\pm}(e, v')$ .

For each possible degree $\mathcal{D}$ , let $\mathcal{E}_{\mathcal{D}}^{\pm}(e) \subseteq \mathcal{E}^{\pm}(e)$ be the set of elevations of degree $\mathcal{D}$ . We will say $\mathcal{X}'$ satisfies Stallings’ condition if and only if the following two things hold:

(a) Every edge map of $\mathcal{X}'$ is an elevation of the appropriate edge map of $\mathcal{X}$ .
(b) For each $e \in E(\Xi)$ and $\mathcal{D}$ , there is a bijection $\mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e)$ .

So in Figure 1, the graph of spaces $\mathcal{X}'$ is something you might be able to turn into a covering. In the picture, $\mathcal{E}_{\mathcal{D}}^+(e)$ is represented by the blue circles, and $\mathcal{E}_{\mathcal{D}}^-(e)$ is represented by the green circles. Observe that the blue circles are in bijection with the green circles.

Corollary: $\mathcal{X}'$ satisfies Stallings’ condition if and only if $\mathcal{X}'$ can be realized as a subgraph of spaces of some $\widehat{\mathcal{X}}$ such that

(a) $V(\Xi ') = V(\widehat{\Xi})$ , and
(b) there is a covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ such that the following diagram commutes:

diagram1

Proof of Corollary. First we’ll show that if $\Phi$ can be extended to a covering map as described above, then $\mathcal{X}'$ satisfies Stallings’ condition. By Theorem 17, every edge map of $\widehat{\mathcal{X}}$ is an elevation. So there are inclusions

diagram2

Furthermore, these maps are surjective, and clearly degree-preserving.

Now assume that $\mathcal{X}'$ satisfies Stallings’ condition. Then we build $\widehat{\mathcal{X}}$ as follows. Let $V(\widehat{\Xi}) = V(\Xi ')$ . As above, we have degree-preserving inclusions (this time, not surjections)

diagram3

Extend these inclusions to bijections $\mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e)$ . Now we set $E(\widehat{\Xi}) = \bigcup_{e \in E(\Xi)} \mathcal{E}^+(e)$ . Each of these $\widehat{e}$ is an elevation

diagram4

This defines an edge space $X_{\widehat{e}}$ and an edge map $\partial_{\widehat{e}}^+$ . Consider the corresponding elevation in $\bigcup_{e \in E(\Xi)} \mathcal{E}^-(e)$ :

diagram5

Because $\partial_{\widehat{e}}^+$ and $\partial_{\widehat{e}}^-$ are of the same degree, we have a covering transformation $X_{\widehat{e}} ' \to X_{\widehat{e}}$ . So we can identify them, and use $\partial_e^-$ as the other edge map. By construction, $\widehat{\mathcal{X}}$ satisfies the conditions of Theorem 17, so there is a suitable covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ .

Exercise 25: (This will be easier later, but we have the tools necessary to do this now.) Prove that if $G_1$ and $G_2$ are LERF groups, then so is $G_1 * G_2$ .

26. Building Covering Spaces

6 April 2009 in Course notes | by krammai | Leave a comment

Lemma 27 Revisited. Suppose $\tau : X' \to X$ is a covering map. Then there is a covering map $\sigma: Y'\to Y$ such that $X'$ is the fibre product of $\sigma$ and $d$ .

Proof. Let $Y' = \left\{ (\xi',\eta) \in X' \times Y : \tau (\xi) = i (\eta) \right\}$ be the fibre product of $\tau$ and $i$ . There is a map $d' : X' \to Y'$ given by $\xi' \mapsto (\xi' , d \circ \tau (\xi'))$ . Let $\hat{X}$ be the fibre product of $\sigma$ and $d$ ; i.e.

$\hat{X} = \left\{ (\xi,\eta') \in X \times Y' : d(\xi) = \sigma(\eta') \right\}$ .

fig1note04011

There is a map $X' \to \hat{X}$ given by $\xi' \mapsto (\tau(\xi') , d'(\xi'))$ . This is a covering map and injective, so it is a homeomorphism.

Let $f: X \to Y$ be continuous, $Y' \to Y$ be a covering map and $x \in X$ , $y= f(x) \in Y$ choices of basepoint. We have already seen that a choice of $y' \in Y'$ such that $y' \mapsto y$ determines an elevation of $f$ to $Y'$ at $y'$ . Fix such a $y'$ . The pre-image of $y$ in $Y'$ is in bijection with the set of cosets

$\pi_1(Y',y') \backslash \pi_1(Y,y)$

This raises the question, when do two cosets determine the same elevation?

Exercise 24. $\pi_1(Y',y')g_1$ and $\pi_1(Y',y')g_2$ determine the same elevation if and only if

$\pi_1(Y',y') g_1 f_* \pi_1(X,x) = \pi_1(Y',y') g_2 f_* \pi_1(X,x)$ ;

that is, the set of elevations of $f$ to $Y'$ is in bijection with $\pi_1(Y') \backslash \pi_1(Y) / f_*(\pi_1(X))$ .

Let $\mathfrak{X}$ and $\mathfrak{X}'$ be graphs of spaces and suppose we have the following data.

(a) A combinatorial map $\Xi' \to \Xi$ given by $e' \mapsto e$ and $v' \mapsto v$ .

(b) Covering maps $\varphi_{v'} : X_{v'} \to X_v$ for each $v' \in V(\Xi')$ .

(c) Covering maps $\varphi_{e'} : X_{e'} \to X_{e}$ for each $e' \in E (\Xi')$ , such that $\varphi_{v'}\circ\partial_{e'} = \partial_e\varphi_{e'}$ whenever $e'$ adjoins $v'$ .

This determines a continuous map $\Phi : X_{\mathfrak{X}'} \to X_{\mathfrak{X}}$ . When is $\Phi$ really a covering map?

Theorem 17. $\Phi$ is a covering map if

(i) for all $e' \in E(\Xi')$ adjoining $v' \in V(\Xi')$ , the edge map $\partial_{e'}^{\pm} : X_{e'} \to X_{v'}$ is an elevation of $\partial_{e}^{\pm} : X_{e} \to X_{v}$ ; and

(ii) wherever $e \in E(\Xi)$ adjoining $v \in V(\Xi)$ and $v' \in V(\Xi')$ , every elevation $\partial_{e}^{\pm} : X_{e} \to X_{v}$ to $X_{v'}$ arises as an edge map of $\mathfrak{X}'$ .

Proof (sketch). It’s enough to consider our local model : $X_{\mathfrak{X}} = X$ and $Y = X_v$ . $\varphi_{v'} : X_{v'} \to X_v$ and $\varphi_{e'} : X_{e'} \to X_{e}$ be covering maps defining $\mathfrak{X}'$ and a map $\Phi : X_{\mathfrak{X}'} \to X_{\mathfrak{X}}$ . By Lemma 27, $\Phi$ is a covering map if and only if $X_{\mathfrak{X}'}$ is a fibre product with respect to some covering: