amalgamated free product
Take two groups and glue them together — but only along a shared piece. Imagine two separate worlds that happen to contain identical copies of one common subgroup; you identify those copies and let everything else mix freely. The amalgamated free product is the most economical group containing both, agreeing on the common part and imposing no further coincidences between the rest.
Given groups A and B and a common subgroup C with injections C to A and C to B, the amalgamated free product A *_C B is the quotient of the ordinary free product A * B by the relations identifying the two images of each c in C. It is precisely the pushout of A and B over C in the category of groups, characterized by the universal property: homomorphisms A *_C B to G correspond to pairs of homomorphisms from A and from B that agree on C.
The structure is controlled by a normal form theorem (Schreier): choosing coset representatives for C in A and in B, every element of A *_C B has a unique expression as c times an alternating product of nontrivial representatives from A and B. A crucial consequence is that A and B both embed into A *_C B (the maps are injective), so the amalgam genuinely contains both factors. When C is trivial this recovers the plain free product A * B.
SL(2, Z) is isomorphic to Z/4Z *_{Z/2Z} Z/6Z: two finite cyclic groups amalgamated over their common central subgroup of order 2. Equivalently PSL(2, Z) is the plain free product Z/2Z * Z/3Z.
SL(2, Z) is an amalgam of two finite cyclic groups along an order-2 subgroup.
By van Kampen's theorem, if a space is the union of two open sets meeting in a connected set, its fundamental group is the amalgam of the pieces' fundamental groups over the intersection's — the topological face of the same construction.