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The free product of finite groups $ A * B $ naturally acts on a biregular graph see Free Product of two finite groups. This seems like one of the only places that free products of finite groups appear naturally in math. I'm curious about other places that free products of finite groups naturally come up. The one other example I can think of is that the arithmetic group $ PSL(2,\mathbb{Z}) $ is a free product of finite (cyclic) groups $ C_2 * C_3 $.

Are there other arithmetic groups that are free products of finite groups? More generally, are there other areas of math where free products of finite groups naturally arise? For example as fundamental groups or mapping class groups of nice spaces (e.g. manifolds or manifolds with boundary)?

Ian Gershon Teixeira
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    I assume you mean outside group theory. Within group theory they are fundamental. Have you come across van Kampen's theorem? – Sean Eberhard Jul 28 '23 at 19:23
  • @SeanEberhard Yes but aren't those usually amalgamated free products? Are there some nice manifolds/ manifolds with boundary that have $ \pi_1 $ a free product of finite groups? – Ian Gershon Teixeira Jul 28 '23 at 19:25
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    If the intersection is simply connected, you get a free product. – Arturo Magidin Jul 28 '23 at 19:33
  • @ArturoMagidin oh ya , so for example could you have a copy of $ \mathbb{R}P^3 $ and a copy of the icosahedral space $ SO_3/A_5 $, which intersect at an $ S^3 $, and the fundamental group would be $ C_2 * SL(2,5) $? – Ian Gershon Teixeira Jul 28 '23 at 19:42
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    I think so, but my algebraic topology is a bit rusty. For not-necessarily-finite groups, you have a bouquet of two circles, each with fundamental group $\mathbb{Z}$, and with intersection a contractible $X$ figure, which gives you the free group of rank $2$. – Arturo Magidin Jul 28 '23 at 19:50
  • wedge sums in spaces give a wealth of such examples! – Andres Mejia Jul 31 '23 at 15:25

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Free products of finite groups are a fundamental class of examples in field of geometric group theory, particularly in the subfield of group actions on trees.

They are usually studied not using manifold theory, but instead using Bass-Serre theory which is the study of graphs of groups and group actions on trees and their inter-relations. The idea of Bass-Serre theory is that every graph of groups has a corresponding "Bass-Serre tree" which is a group acting on a tree that plays a role quite like the fundamental group acting on the universal covering space.

In the realm of Bass-Serre theory, free products of finite groups come up naturally as simple examples of fundamental groups of graphs of groups with trivial edge groups and finite vertex groups. Serre's beatiful book "Trees" contains more examples and the elements of the theory.

Free products of finite groups, taken together with ordinary finite rank free groups, also occupy an important position on the current frontier of geometric group theory: given a group $\Gamma$ in this class, the outer automorphism group $\text{Out}(\Gamma)$ and its action on the deformation spaces of $\Gamma$-actions on trees are the focus of a lot of current research. The elevator pitch for this branch of mathematical research is that it aims to discover and prove theorems by analogy with mapping class groups of surfaces, which act on the deformation space of hyperbolic structures on surfaces (i.e. on Teichmuller spaces).

Lee Mosher
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