Let $$S_g = \langle a_1,b_1,...,a_g,b_g \mid \prod_{i=1}^g[a_i,b_i] = 1 \rangle$$ be the fundamental group of a genus $g$ orientable surface. Why is $S_g \ast S_h \cong S_{g+h}$, and is there a nice canonical isomorphism?
2026-03-29 15:54:36.1774799676
Free product of surface groups
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Those groups are not isomorphic, for example the free product has infinitely many ends and the surface group has one end since it is quasi-isometric to the hyperbolic plane (surface groups act geometrically on the hyperbolic plane you can apply Svarc-Milnor) which is one ended. You might be interested in Stallings theorem on ends.