Commutator of free products of groups

Question

Let $G$ be a free product of nontrivial groups $H_i$: $G=H_1\ast\cdots\ast H_n.$

Let $g_1,g_2\in G$ be such elements that: 1) $g_1,g_2\not\in H_k^g$ for all $g\in G$ and $k=1,\ \dots,\ n$; 2) $[g_1,g_2]=g_1g_2g_1^{-1}g_2^{-1}\ne 1$.

Is it true that $[g_1,g_2] \not\in H_k^g$ for all $g\in G$ and $k=1,\ \dots,\ n$?

Answer 1

Here is a sketch of proof:

• $G$ acts on its Bass-Serre tree $T$. Since the vertex-stabilisers are precisely the conjugates of the free factors, your assumption 1 precisely means that $g_1$ and $g_2$ are loxodromic isometries of $T$.

• For $i=1,2$, let $\ell_i$ denote the axis of $g_i$. Since $g_1$ and $g_2$ do not commute, $\ell_1 \neq \ell_2$.

• Prove that the commutator $[g_1,g_2]$ is also a loxodromic element, meaning that $[g_1,g_2]$ does not belong to the conjugate of some free factor.