Graphs and Free Products

Let $\Gamma = \Gamma(E, V)$ be a finite graph and let $X = |\Gamma|$ be its geometric realization.
- (i) Show that $X$ is connected if and only if for each pair of distinct vertices $v_1$ and $v_2$ of $\Gamma$, there is an edge path from $v_1$ to $v_2$.
- (ii) Let $\hat{X}$ be a finite-sheeted cover of $X$. Show that there exists a graph $\hat{\Gamma}$ such that $\hat{X} = |\hat{\Gamma}|$.
For a set $S$, let $\langle S \rangle$ denote the free group generated by $S$.
- (i) If $S=\phi$, show that $\langle S \rangle$ is the trivial group.
- (ii) If $S$ is a singleton set, show that $\langle S \rangle$ is isomorphic to $\mathbb{Z}$.
- (iii) Let $\{S_\alpha\}_{\alpha\in A}$ be a collection of disjoint sets. Show that the free product $\star_{\alpha\in A} \langle S_\alpha \rangle$ is isomorphic to the free group $\langle \cup_{\alpha\in A} S_\alpha \rangle$ .