Hatcher-Algebraic-Topology · Problem 1.3.30

The key is to understand that this group is the free product of $\mathbb{Z}$ (infinite cyclic, generated by $a$) and $\mathbb{Z}_2$ (generated by $b$ with $b^2 = e$). Every element is a reduced word alternating between powers of $a$ and $b$. The Cayley graph shows how generators $a$ and $b$ connect these words: $a$-edges extend or reduce powers of $a$, while $b$-edges flip the parity of $b$ in the word. The result is an infinite tree-like structure with a branching pattern that reflects the interplay between the two generators.

The key is to understand that this group is the free product of $\mathbb{Z}$ (infinite cyclic, generated by $a$) and $\mathbb{Z}_2$ (generated by $b$ with $b^2 = e$). Every element is a reduced word alternating between powers of $a$ and $b$. The Cayley graph shows how generators $a$ and $b$ connect these words: $a$-edges extend or reduce powers of $a$, while $b$-edges flip the parity of $b$ in the word. The result is an infinite tree-like structure with a branching pattern that reflects the interplay between the two generators.