Algebraic and Geometric intersection numbers for free groups

TL;DR

This paper demonstrates that the algebraic intersection number for free group splittings matches the geometric intersection number in the sphere complex of connected sums of S^2×S^1, linking algebraic and geometric perspectives.

Contribution

It establishes the equivalence between algebraic and geometric intersection numbers for free groups, bridging two different approaches in the field.

Findings

Algebraic and geometric intersection numbers coincide for free group splittings.

The result connects algebraic properties of free groups with geometric structures.

Provides a unified understanding of intersection concepts in free group theory.

Abstract

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.