A nice example is the fundamental group of the torus. View it as a rectangle with opposite sides identified. Let U be the interior of the rectangle, and let V be a neighborhood of the boundary of the rectangle. pi(U) is trivial. V deformation retracts to the boundary, which is the one-point union of two circles. It has fundamental group Z*Z (by Van Kampen's theorem!). U intersect V is an annulus, with fundamental group Z. Now Van Kampen's theorem says the torus is the product of Z*Z with the trivial group amalgamated over the image of the fundamental group of the annulus. If a and b are the generators of Z*Z, then the image of the annulus is generated by aba^-1b^-1, the commutator of a and b. So the torus has fundamental group presentation , which is just the free abelian group on two generators.

A nice example is the fundamental group of the torus. View it as a rectangle with opposite sides identified. Let U be the interior of the rectangle, and let V be a neighborhood of the boundary of the rectangle. pi(U) is trivial. V deformation retracts to the boundary, which is the one-point union of two circles. It has fundamental group Z*Z (by Van Kampen's theorem!). U intersect V is an annulus, with fundamental group Z. Now Van Kampen's theorem says the torus is the product of Z*Z with the trivial group amalgamated over the image of the fundamental group of the annulus. If a and b are the generators of Z*Z, then the image of the annulus is generated by aba^-1b^-1, the commutator of a and b. So the torus has fundamental group presentation , which is just the free abelian group on two generators.