[Paper Review] Makanin-Razborov Diagrams over Free Products

Éric Jaligot, Zlil Sela|ArXiv.org|Feb 14, 2009

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper extends the Makanin-Razborov diagram framework to finitely presented groups over free products, introducing limit groups over free products with elliptic elements and constructing a non-canonical diagram encoding all free product quotients. The key contribution is a uniform diagram that captures all homomorphisms into arbitrary free products, using a derived d.c.c. on proper epimorphisms and resolution constructions, though finiteness of maximal quotients remains open for finitely generated groups.

ABSTRACT

This paper is the first in a sequence on the first order theory of free products. In the first paper we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze system of equations over free products. To do that we introduce limit groups over the class of free products, and show that a finitely presented group has a canonical (finite) collection of maximal limit quotients. We further extend this finite collection and associate a Makanin-Razborov diagram over free products with a finitely presented group. This MR diagram encodes all the quotients of a given finitely presented group that are free products, all its homomorphisms into free products, and equivalently all the solutions to a given system of equations over a free product.

Motivation & Objective

• To generalize the Makanin-Razborov diagram framework from free groups to arbitrary free products.
• To define and study limit groups over free products, incorporating a new structure: conjugacy classes of elliptic elements.
• To construct a diagram encoding all free product quotients of a finitely presented group.
• To establish a descending chain condition (d.c.c.) on proper epimorphisms that preserve non-trivial elliptic elements.
• To explore the possibility of a canonical diagram and identify open problems for finitely generated groups.

Proposed method

• Define limit groups over free products as quotients of a finitely presented group by the stable kernel of a convergent sequence of homomorphisms into free products.
• Introduce a canonical collection of conjugacy classes of elliptic elements in limit groups, which map to conjugates of the factors in the target free product.
• Construct a virtually abelian JSJ decomposition for each limit group over free products.
• Use a descending chain condition on proper epimorphisms that do not map non-trivial elliptic elements to identity to build finite resolutions.
• Associate each limit group with a resolution, enabling the construction of a uniform Makanin-Razborov diagram over free products.
• Define a partial order on homomorphisms into free products to explore maximality and potential construction of a canonical diagram.

Experimental results

Research questions

• RQ1 Can the Makanin-Razborov diagram framework be extended to finitely presented groups over arbitrary free products?
• RQ2 How should limit groups over free products be defined, and what additional structure (e.g., elliptic elements) is necessary to capture homomorphisms into free products?
• RQ3 Does a descending chain condition hold for limit groups over free products under proper epimorphisms that preserve non-trivial elliptic elements?
• RQ4 Can a canonical Makanin-Razborov diagram be constructed for finitely presented groups over free products using maximal homomorphisms?
• RQ5 Are there finitely many equivalence classes of maximal limit quotients over free products for a finitely generated group?

Key findings

• A uniform Makanin-Razborov diagram is constructed for finitely presented groups over free products, encoding all homomorphisms into arbitrary free products.
• The diagram is not canonical due to reliance on non-unique finite covers in the resolution construction.
• A descending chain condition (d.c.c.) is established for limit groups over free products under proper epimorphisms that preserve non-trivial elliptic elements.
• Each limit group over free products admits a canonical virtually abelian JSJ decomposition.
• The existence of finitely many maximal limit quotients over free products for a finitely presented group is established, but the finiteness for finitely generated groups remains open.
• A natural conjecture is formulated: every increasing sequence of homomorphisms into free products stabilizes, which would enable a canonical diagram construction.