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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 Mis 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

  1. P_i/N_i\to G/\mathcal{N} is injective for all i, and
  2. 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.

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.

  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,

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\}\ranglet is called a stable letter.

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