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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