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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!

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  • $\begingroup$ What is your background? How much geometric group theory do you know? Both claims are indeed trivial if you have a sufficient background. $\endgroup$ Nov 2, 2023 at 13:11
  • $\begingroup$ Please ask one question at a time. $\endgroup$
    – Shaun
    Nov 2, 2023 at 13:40
  • $\begingroup$ 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, ... $\endgroup$
    – Mergolyx
    Nov 2, 2023 at 13:56
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    $\begingroup$ 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? $\endgroup$ Nov 2, 2023 at 21:57
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    $\begingroup$ You can start by reading here. $\endgroup$ Nov 3, 2023 at 1:32

1 Answer 1

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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...

(The free product case was already addressed in comments, using that negatively curved surface groups don't split into nontrivial free products.)

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