In this lecture, we will use the Normal Form Theorem to understand and construct groups.  In particular, we will construct a group that is not residually finite (RF).  Note that in the case when $\mathcal{G}$ is an HNN-extension, then the Normal Formal Theorem is called Britton’s Lemma.

Example: The (p,q)-Baumslag-Solitar group has the following presentation:

For example $\mathrm{BS} (1,1) \cong \mathbb{Z}^{2}$.

Consider $\mathrm{BS}(2,3)$.  We will show that this group is non-Hopfian (i.e., that there exists a homomorphism $\varphi: \mathrm{BS}(2,3) \to \mathrm{BS}(2,3)$ that is a surjection with nontrivial kernel), hence is not RF.  Consider $\psi: \mathrm{BS}(2,3) \to \mathrm{BS}(2,3)$ defined by $\psi(t) = t$ and $\psi(a) = a^{2}$.  First, we show this is a homomorphism, then a surjection, then that it has nontrivial kernel.

Homomorphism:  $\psi(ta^{2}t^{-1}) = ta^{4}t^{-1} = ta^{2}t^{-1}ta^{2}t^{-1} = a^{3}a^{3} = a^{6} = \psi(a^{3}).$  Hence, the map is a homomorphism.

Epimorphism: To see that $\psi$ is a surjection, we check that $a, t \in \mathrm{Im}(\psi)$.  This is obvious for $t$.  Further $a^{2} = \psi(a) \in \mathrm{Im}(\psi)$ and so $ta^{2}t^{-1} = a^{3} \in \mathrm{Im}(\psi)$.  But then $a = a^{3}a^{-2} \in \mathrm{Im}(\psi)$.

Nontrivial kernel: Consider $c = ata^{-1}t^{-1}a^{-2}tat^{-1}a$.  Then $\psi(c) = a^{2}ta^{-2}t^{-1}a^{-4}ta^{2}t^{-1}a^{2} = a^{2}a^{-3}a^{-4}a^{3}a^{2} = 1$.  So we just need to show that $c$ is nontrivial.  We use Britton’s Lemma (Normal Form Theorem).  What Britton’s Lemma says is that there will be $t's$ that can be removed from our word using the relation if it is trivial.  But this can’t be done with $c$, so we must have $c \neq 1$ as required.

Remark: $\mathrm{BS}(2,3)$ can never embed in a word-hyperbolic group by a previous Lemma.

Exercise 20: Show that $\mathrm{BS}(1,2) \cong \mathbb{Q}_{2} \rtimes_{2} \mathbb{Z}$, where $\mathbb{Q}_{2}$ is the dyadic rationals and we have $\mathbb{Z}$ acting via multiplication by 2.  One can deduce that $\mathrm{BS}(1,2)$ is linear, hence RF by Selberg’s Lemma.

Theorem 15 (Higman-Neumann-Neumann): Every countable group embeds in a 3-generator group.

Proof: Let $G = \{ 1 = g_{0}, g_{1}, \ldots\}$.  Consider $G \ast \mathbb{Z} = G \ast \langle s \rangle$.  This has some nice free subgroups, unlike $G$:  Let $s_{n} = g_{n}s^{n}$.  Let $\Sigma_{1} = \{ s_{n} \vert n \geq 1\}$.  By the Normal Form Theorem, $\langle \Sigma_{1} \rangle \cong F_{\Sigma_{1}}$ (basically since there will always be some g’s between the s’s whenever you multiply any two elements together).  Let $\Sigma_{2} = \{ s_{n} \vert n \geq 2\}$.  Again, $\langle \Sigma_{2} \rangle \cong F_{\Sigma_{2}}$.  Since $latex\Sigma_{1}$ and $\Sigma_{2}$ are both countable, $F_{\Sigma_{1}} \cong F_{\Sigma_{2}}$.

Consider the HNN-extension $\Gamma = (G \ast \langle s \rangle)\ast_{F_{\Sigma_{1}} \tilde F_{\Sigma_{2}}}$ and let t be the stable letter.  For every $n \geq 1$, $ts_{n}t^{-1} = s_{n+1}$.  But $\Gamma$ is generated by $g_{1}$, $s$, and $t$: $s_{n+1} = ts_{n}t^{-1}$, so $g_{n+1}s^{n+1} = tg_{n}s^{n}t^{-1}$ or $g_{n+1} = tg_{n}s^{n}t^{-1}s^{-n-1}$.  So by induction $g_{n} \in \langle g_{1}, s, t \rangle$. for all n.  But by construction $\Gamma$ is generated by $\Sigma_{1} \cup \{ s\} \cup \{ t \}$. $\square$

Remark: In fact 3 can be replaced by 2.

Isometries of Trees

Before we saw that graphs of groups correspond to groups acting on trees.  As such, we now turn to isometries of trees.  Let $T$ be a tree and let $\gamma \in \mathrm{Isom}(T)$.  The translation length of $\gamma$ is defined to be $|\gamma| := \inf_{x \in T} d(x, \gamma x)$.  Let $\mathrm{Min}(\gamma) = \{ x \in T \vert d(x, \gamma x) = |\gamma| \}$.

Definition: If $\mathrm{Min}(\gamma) \neq \emptyset$, $\gamma$ is called semisimple.  If $\gamma$ is semisimple, and $|\gamma| = 0$, $\gamma$ is called elliptic.  If $|\gamma| > 0$, it is called loxodromic.

Theorem 16: Every $\gamma \in \mathrm{Isom}(T)$ is semisimple.  If $\gamma$ is loxodromic, $\mathrm{Min}(\gamma)$ is isometric to $\mathbb{R}$, and $\gamma$ acts on $\mathrm{Min}(\gamma)$ as translation by $|\gamma|$.

Notation: If $\gamma$ is elliptic, we write $\mathrm{Min}(\gamma) = \mathrm{Fix}(\gamma)$.  If $\gamma$ is loxodromic\$, we write $\mathrm{Min}(\gamma) = \mathrm{Axis}(\gamma)$.  Note that $\mathrm{Fix}(\gamma)$ is connected since if we have two points fixed by $\gamma$ any path between them must be fixed, for otherwise we would not be in a tree.

Case 1 of Theorem 16

Proof of Theorem 16: Consider any $x \in T$ and the triangle with vertices $x, \gamma x, \gamma^{2} x$ (i.e., $[x, \gamma x] \cup \gamma [x, \gamma x]$).  Let $[ x, \gamma x] \cap [ \gamma x, \gamma^{2} x ] \cap [ \gamma^{2} x, x ] = \{ O \}$, and let $M$ be the midpoint of $[ x, \gamma x ]$.

Case 1: $d(M, \gamma x) \leq d(O,\gamma x)$

In this case we have $d(M, \gamma x) = d(M, x) = d(\gamma M, \gamma x)$.  So $\gamma M = M$, and $\gamma$ is elliptic.

Case 2: $d(M, \gamma x) > d(O, \gamma x)$

Case 2 of Theorem 16

Let $I = [\gamma^{-1} O, O] \subseteq [x, \gamma x]$.  Now $I \cap \gamma I = \{ O \}$.  Therefore $\bigcup\limits_{n \in \mathbb{Z}} \gamma^{n} I$ is isometric to $\mathbb{R}$ and $\gamma$ acts as translation by $[\gamma^{-1}O, O]$. Furthermore, $d(x, \gamma x) = d(\gamma^{-1}O, O) + 2 d(O, \gamma x)$.  Therefore unless $x$ is on the line just constructed, $d(x, \gamma x) > d(\gamma^{-1} O, O) = |\gamma|$. $\square$

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