# Hyperbolic surface subgroups of products of groups

I am an undergraduate student, currently working in an REU project about geometric group theory. I know a few basic notions of geometric group theory and algebraic topology: Cayley graphs, fundamental groups (of topological spaces and of graphs of groups), Seifert-van Kampen theorem, classification of compact 2-manifolds, ...

While reading the following article: Surface subgroups of Coxeter and Artin groups; I encountered the following questions.

Question.

1. If the direct product $$A\times B$$ of two groups $$A$$, $$B$$ contains a hyperbolic surface subgroup, then either $$A$$ or $$B$$ contains a hyperbolic surface subgroup.
2. If the free product $$A*B$$ of two groups $$A$$, $$B$$ contains a hyperbolic surface subgroup, then either $$A$$ or $$B$$ contains a hyperbolic surface subgroup.

Here, a hyperbolic surface group means a fundamental group of a closed, compact $$2$$-manifold with genus $$g >1$$, which can be presented as $$\langle a_1, b_1, \ldots, a_g, b_g \hspace{0.1cm}|\hspace{0.1cm} [a_1, b_1][a_2, b_2]\cdots [a_g, b_g] = 1\rangle$$.

The article that I am reading claims that this is "trivial", however, I do not know how to prove these results. Please give me hints or ideas to prove the above claims. Thank you in advance!

• What is your background? How much geometric group theory do you know? Both claims are indeed trivial if you have a sufficient background. Nov 2, 2023 at 13:11
• Please ask one question at a time. Nov 2, 2023 at 13:40
• Dear Kohan, I am an undergraduate student, currently working in a REU project about geometric group theory. I know a few basic notions of geometric group theory and algebraic topology: Cayley graphs, fundamental groups (of topological spaces and of graphs of groups), Seifert-van Kampen theorem, classification of compact 2-manifolds, ... Nov 2, 2023 at 13:56
• OK, do you know that fundamental groups of surfaces do not split as nontrivial free products? Do you know that two elements of a hyperbolic surface group commute iff they generate a cyclic subgroup? Nov 2, 2023 at 21:57
• You can start by reading here. Nov 3, 2023 at 1:32

Let $$G$$ be a group with the property that the centralizer of every conjugacy class $$\neq\{1\}$$ is trivial (example: $$G$$ negatively curved surface group).
Then $$G$$ has the following property: whenever $$G\subset A\times B$$, $$G$$ projects injectively into either $$A$$ or $$B$$ (indeed the kernels of two projections are normal subgroups of $$G$$ centralizing each other...