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The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation $$ \langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle. $$ The proof I know ...

Good afternoon everyone, I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...