Amalgamated Free Products

Algebraic Topology · Axiom Academy

LESSON Amalgamated Free Products Free products with identified common subgroups 1. Motivation: Shared Subgroup Suppose we have groups G and H, each containing a common subgroup K via injective homomorphisms i: K → G and j: K → H. The ordinary free product G * H treats elements from K ⊆ G and K ⊆ H as distinct. Problem: In G * H, the element i(k) from G and j(k) from H are different, even though they both represent the same k ∈ K. Solution: The amalgamated product G *_K H identifies these elements by imposing the relation i(k) = j(k) for all k ∈ K. The amalgamated product is constructed by starting with the free product G * H and then quotienting out by the normal subgroup generated by elements of the form i(k)j(k)⁻¹ for all k ∈ K. Let N be the normal subgroup of G * H generated by i(k)j(k)⁻¹ : k ∈ K . Then: If we have presentations G = ⟨S_G | R_G⟩, H = ⟨S_H | R_H⟩, and K = ⟨S_K | R_K⟩, we can construct a presentation for the amalgamated product. The amalgamated product has generators from both G and H, relations from both, plus additional relations identifying elements of K: Relations: R_G ∪ R_H ∪ i(k) = j(k) : k ∈ K Amalgamated products appear throughout topology and group theory. Example 1: If K = e , then G *_K H = G * H (the ordinary free product). Example 2: If G = H = ℤ and K = 2ℤ, then G *_K H ≅ ℤ (the integers). Example 3: The trefoil knot group can be expressed as ℤ/3ℤ *_ ℤ/2ℤ ℤ/3ℤ where the shared ℤ/2ℤ comes from a meridian.

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