几何群论笔记

几何群论笔记（二）：融合积与正规形

1 Direct limits

Let \(I\) be a set. For each \(i\in I\), let \(G_i\) be a group. For \(i,j\in I\), let \(\mathcal F_{ij}\) be a possibly empty set of homomorphisms from \(G_i\) to \(G_j\).

Definition. The direct limit of the \(G_i\) relative to the \(\mathcal F_{ij}\), denoted by

\[ G=\varinjlim G_i, \]

is a group together with homomorphisms \(\iota _i:G_i\to G\) satisfying the following universal property:

\[ \iota _j\circ f=\iota _i\qquad (f\in \mathcal F_{ij}). \]

For every group \(H\) and every family \(h_i:G_i\to H\) such that \(h_j\circ f=h_i\) for all \(f\in \mathcal F_{ij}\), there exists a unique homomorphism

\[ h:G\longrightarrow H,\qquad h_i=h\circ \iota _i. \]

Figure 1: The universal property of the direct limit.

Example. Take \(G_0,G_1,G_2\), with

\[ \mathcal F_{01}=\{f_1:G_0\to G_1\},\qquad \mathcal F_{02}=\{f_2:G_0\to G_2\}. \]

Then \(\varinjlim G_i\) is the pushout of the diagram

\[ G_1\xleftarrow {\ f_1\ }G_0\xrightarrow {\ f_2\ }G_2. \]

Proposition. The direct limit \(\varinjlim G_i\) exists and is unique up to isomorphism.

Proof. Uniqueness follows from the universal property. For existence, let \(G_i=\langle X_i\mid R_i\rangle \). Write

\[ *_{i\in I}G_i =\left \langle \coprod _{i\in I}X_i\ \middle |\ \coprod _{i\in I}R_i\right \rangle . \]

Let \(j_i:G_i\to *_{i\in I}G_i\) be the canonical homomorphisms and define

\[ G=\left \langle *_{i\in I}G_i\ \middle |\ j_i(g)=j_j(f(g)),\quad f\in \mathcal F_{ij},\ g\in G_i\right \rangle . \]

Suppose \((H,h_i)\) satisfies \(h_i=h_j\circ f\) for every \(f\in \mathcal F_{ij}\). The homomorphism on the free product determined by the \(h_i\) respects these relations, so it induces

\[ h:G\longrightarrow H,\qquad h\bigl (\iota _{i_1}(g_1)\cdots \iota _{i_n}(g_n)\bigr ) =h_{i_1}(g_1)\cdots h_{i_n}(g_n). \]

This is the required homomorphism.

2 Amalgamated products

For homomorphisms \(G_1\xleftarrow {f_1}G_0\xrightarrow {f_2}G_2\), the group

\[ G_1*_{G_0}G_2 =\left \langle G_1*G_2\ \middle |\ f_1(g_0)=f_2(g_0),\ g_0\in G_0\right \rangle \]

is called the amalgamated product of \(G_1\) and \(G_2\) along \(G_0\).

We will assume that the maps into the factors are injective. Otherwise the resulting group might be trivial. For example, let \(A,B\) be finite simple groups, and choose \(1\ne a\in A\), \(1\ne b\in B\) with

\[ \gcd \bigl (\operatorname {ord}(a),\operatorname {ord}(b)\bigr )=1. \]

Let \(f_1:\mathbb Z\to A\), \(1\mapsto a\), and \(f_2:\mathbb Z\to B\), \(1\mapsto b\). Then \(A*_{\mathbb Z}B=\{1\}\).

Application: van Kampen. Let \(X=U_1\cup U_2\), where \(U_1,U_2\) are open and \(U_1,U_2,U_1\cap U_2\) are path connected. Choose a base point \(x\in U_1\cap U_2\). Then

\[ \pi _1(X,x)\cong \pi _1(U_1,x)*_{\pi _1(U_1\cap U_2,x)}\pi _1(U_2,x). \]

Definition. Let \(A\) and \(G_i\), \(i\in I\), be groups, and let \(f_i:A\to G_i\) be injective homomorphisms. Define

\[ *_{A,\,i\in I}G_i=\varinjlim (A,G_i;f_i), \]

the amalgamated product of the \(G_i\) along \(A\). When \(A=\{1\}\), this is the free product \(*_{i\in I}G_i\).

3 Reduced words

For each \(i\), identify \(A\) with \(f_i(A)\subset G_i\). Let \(S_i\) be a set of representatives of the right cosets \(A\backslash G_i\), with \(1\in S_i\). Then

\[ A\times S_i\longrightarrow G_i,\qquad (a,s)\longmapsto as \]

is a bijection. Furthermore,

\[ A\times (S_i\setminus \{1\})\longrightarrow G_i\setminus A \]

is a bijection.

Let \(\mathbf i=(i_1,\ldots ,i_n)\) satisfy

\begin{equation} i_j\ne i_{j+1}\qquad (1\le j<n). \label {eq:ggt2-type-condition} \end{equation}

A reduced word of type \(\mathbf i\) is a family

\[ (a;s_1,\ldots ,s_n),\qquad a\in A,\quad s_j\in S_{i_j}\setminus \{1\}. \]

The empty type \(\mathbf i=\varnothing \) is allowed, giving the words \((a;)\). Write \(G=*_{A,\,i\in I}G_i\). Let \(\iota :A\to G\) and \(\iota _i:G_i\to G\) be the canonical homomorphisms.

Theorem 1. Every \(g\in G\) has a unique expression

\[ g=\iota (a)\,\iota _{i_1}(s_1)\cdots \iota _{i_n}(s_n) \]

as a reduced word \((a;s_1,\ldots ,s_n)\) of a type satisfying (1).

Proof sketch. For existence, write \(g=g_1\cdots g_n\), \(g_j\in G_{i_j}\). Combine adjacent factors from the same group and absorb factors in \(A\). Decomposing factors from the right gives

\[ \begin {aligned} g_1\cdots g_n &=g_1\cdots (g_{n-1}a_n)s_n\\ &=g_1\cdots g_{n-2}a_{n-1}s_{n-1}s_n\\ &\ \vdots \\ &=as_1\cdots s_n. \end {aligned} \]

Remove representatives equal to \(1\) and combine adjacent factors when necessary. Thus every element has such an expression.

For uniqueness, let \(X_{\mathbf i}\) be the set of reduced words of type \(\mathbf i\), and put

\[ X=\coprod _{\mathbf i}X_{\mathbf i}. \]

Define an action of \(G_i\) on \(X\). For \(g_i\in G_i\) and \(m=(a;s_1,\ldots ,s_n)\), use the decomposition \(G_i=A S_i\):

\[ g_i\cdot m= \begin {cases} (a';s',s_2,\ldots ,s_n),&i=i_1,\quad g_i a s_1=a's',\\ (a';s_0',s_1,\ldots ,s_n),&i\ne i_1,\quad g_i a=a's_0'. \end {cases} \]

Omit \(s'\) or \(s_0'\) when it equals \(1\). For the empty word, decompose \(g_i a=a's_0'\) in the same way.

These actions agree on \(A\), so the universal property gives an action of \(G\) on \(X\). Let \(\mathbf 1=(1;)\). If \(g=as_1\cdots s_n\), then

\[ g\cdot \mathbf 1=(a;s_1,\ldots ,s_n). \]

Define

\[ \alpha :G\to X,\quad g\mapsto g\cdot \mathbf 1, \qquad \beta :X\to G,\quad (a;s_1,\ldots ,s_n)\mapsto as_1\cdots s_n. \]

Then \(\alpha \circ \beta =\operatorname {id}_X\), so \(\beta \) is injective. This proves uniqueness.

4 A formulation without coset representatives

For each \(i\in I\), let \(G_i'=G_i\setminus A\). For a nonempty type \(\mathbf i=(i_1,\ldots ,i_n)\) satisfying (1), let \(G_{\mathbf i}'\) be the quotient of \(G_{i_1}'\times \cdots \times G_{i_n}'\) by the action of \(A^{n-1}\) given by

\[ \begin {aligned} &(a_1,\ldots ,a_{n-1})\cdot (g_1,\ldots ,g_n)\\ &\qquad =(g_1a_1^{-1},\ a_1g_2a_2^{-1},\ \ldots ,\ a_{n-1}g_n). \end {aligned} \]

The map

\[ (g_1,\ldots ,g_n)\longmapsto \iota _{i_1}(g_1)\cdots \iota _{i_n}(g_n) \]

induces a map \(\iota _{\mathbf i}:G_{\mathbf i}'\to G\).

Theorem 2. The maps \(\iota \) and \(\iota _{\mathbf i}\) define a bijection

\[ A\ \sqcup \!\coprod _{\mathbf i\ne \varnothing }G_{\mathbf i}' \longrightarrow G. \]

Corollary. If

\[ g=ag_1\cdots g_n,\qquad a\in A,\quad g_j\in G_{i_j}\setminus A,\quad i_j\ne i_{j+1}, \]

then \(g=1\) if and only if \(a=1\) and \(n=0\).

5 Consequences of the structure theorem

For \(G=*_{A,\,i\in I}G_i\), the type of \(g\in G\) is the sequence \(\mathbf i=(i_1,\ldots ,i_n)\) in its reduced expression

\[ g=as_1\cdots s_n,\qquad s_j\in S_{i_j}\setminus \{1\}. \]

The type is empty if and only if \(g\in A\), and it does not depend on the choice of the \(S_i\). Indeed, it is characterized by the factors \(G_{i_j}'\) occurring in Theorem 2. Define the length of \(g\) by \(\ell (g)=n\).

An element \(g\) is called cyclically reduced if its type has \(n\ge 2\) and \(i_1\ne i_n\).

Proposition.

Every element of \(G\) is conjugate to a cyclically reduced element or to an element of one of the \(G_i\).
Every cyclically reduced element has infinite order.

阅读导航上一篇：概览及群论回顾；系列目录。