Problem Sets

Problem Set 1 (due March 7)

Let $G$ be a finitely generated group with a finite generating set $S$ and let $\gamma$ be its growth function with respect to $S$.
- Show that for all $a,b\ge 0$, $\gamma(a+b)\le \gamma(a)\gamma(b)$.
- Show that the growth exponent $e_S=\lim_{n\to \infty}\frac{\log \gamma(n)}{n}$ exists.
Let $G$ be a group, let $A,A'$ be subgroups of $G$, and let $\phi\from A\to A'$ be an isomorphism. For any $g\in G$, let $\phi^g\from A\to gA'g^{-1}$ be the map $\phi^g(x)=gxg^{-1}$. Show that the HNN extensions $G\ast_{\phi}$ and $G\ast_{\phi^g}$ are isomorphic.
Recall the description of $BS(1,2)$ as an HNN extension $BS(1,2)\cong \Z\ast_\phi$ from class, where $\phi\from \Z\to 2\Z$ is multiplication by 2. Describe the corresponding tree $T$ and the action of $BS(1,2)$ on $T$.
(Britton's Lemma) Let $$H=G\ast_{\phi}=\langle G,t\mid t a t^{-1}=\phi(a) \forall a\in A\rangle$$ be an HNN extension, where $\phi\from A\to B$ is an isomorphism between subgroups of $G$. Any element $h\in H$ can be written as a product $$h=g_0 t^{\epsilon_1}g_1t^{\epsilon_2} \dots g_{n-1}t^{\epsilon_{n}}g_n,$$ where $g_i\in G$ and $\epsilon_i\in \{-1,1\}$. Show that if $h=1$ and $n>0$, then the product contains a substring of the form $tat^{-1}$ or $t^{-1}bt$ for some $a\in A$ or $b\in B$.