[Paper Review] Intersecting free subgroups in free amalgamated products of two groups with normal finite amalgamated subgroup

Alexander Zakharov|arXiv (Cornell University)|May 12, 2011

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper generalizes the Hanna Neumann inequality to free amalgamated products of groups with a normal finite amalgamated subgroup. By analyzing the structure of factor-free subgroups via group factorization and graph-theoretic methods, it establishes a sharp upper bound on the reduced rank of the intersection of two such subgroups, incorporating the order of the amalgamated subgroup and the minimal order >2 of subgroups in the quotient groups. The bound is shown to be unimprovable when the quotient groups contain involutions and are not isomorphic to ℤ₂⁎ℤ₂.

ABSTRACT

We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We also prove that the obtained estimate is sharp and cannot be further improved when the amalgamated product contains an involution.

Motivation & Objective

• To extend the Hanna Neumann inequality to free amalgamated products where the amalgamated subgroup is finite and normal.
• To establish a sharp upper bound on the reduced rank of the intersection of two factor-free subgroups in such amalgamated products.
• To determine conditions under which the derived bound is unimprovable, particularly when the quotient groups contain involutions.

Proposed method

• Use of group factorization φ: G₁⁎ₜG₂ → G₁/T⁎G₂/T to relate subgroups in the amalgamated product to subgroups in the free product of quotients.
• Application of Theorem 1 (Dicks and Ivanov) on reduced rank bounds in free products to the quotient group G₁/T⁎G₂/T.
• Employment of the Schreier formula to relate the reduced rank of the intersection to the index of subgroups in the quotient group.
• Use of graph-theoretic methods (Ψ* graphs) to analyze subgroup structure and prove sharpness of the bound.
• Construction of explicit examples in cases where G₁/T or G₂/T contains an involution and G₁/T⁎G₂/T ≇ ℤ₂⁎ℤ₂ to demonstrate unimprovability.
• Verification that the index |H₁′ ∩ H₂′ : L| is bounded by |T|, leading to the final rank inequality.

Experimental results

Research questions

• RQ1 Can the Hanna Neumann inequality be generalized to free amalgamated products with a normal finite amalgamated subgroup?
• RQ2 What is the sharp upper bound on the reduced rank of the intersection of two factor-free subgroups in such amalgamated products?
• RQ3 Under what conditions is the derived bound unimprovable?
• RQ4 How does the order of the amalgamated subgroup |T| and the minimal order >2 of subgroups in G₁/T and G₂/T affect the rank bound?
• RQ5 Is the bound sharp when the quotient groups contain involutions and are not isomorphic to ℤ₂⁎ℤ₂?

Key findings

• The reduced rank of the intersection of two factor-free subgroups H₁ and H₂ in a free amalgamated product G₁⁎ₜG₂ with normal finite T satisfies r̄(H₁ ∩ H₂) ≤ 2(q_f^*/(q_f^*−2))|T| r̄(H₁)r̄(H₂).
• The bound is sharp and cannot be improved when G₁/T or G₂/T contains an involution and G₁/T⁎G₂/T ≇ ℤ₂⁎ℤ₂.
• The index |H₁′ ∩ H₂′ : L| is bounded above by |T|, where H₁′, H₂′ are the images of H₁, H₂ under the factor map to G₁/T⁎G₂/T.
• The bound reduces to the classical Hanna Neumann inequality when T is trivial (i.e., G₁⁎G₂ is a free product).
• The sharpness of the bound is demonstrated via explicit constructions in four cases, including when G₁/T⁎G₂/T ≅ ℤ₂⁎ℤ₂⁎ℤ₂.
• The result generalizes earlier work by Dicks and Ivanov on free products and extends the scope of rank inequalities to a broader class of amalgamated products.