# The contraction category of graphs

## TL;DR

This paper investigates a category of graphs with fixed genus and contractions, demonstrating that associated module categories are Noetherian and finitely generated, leading to insights on homology, Betti numbers, and Kazhdan-Lusztig coefficients.

## Contribution

It introduces a new categorical framework for graphs of fixed genus and proves finiteness properties of modules related to graph homology and intersection homology.

## Key findings

- Modules over the graph contraction category are finitely generated.
- Finiteness results imply bounds on torsion in homology groups.
- Growth rates of Betti numbers and Kazhdan-Lusztig coefficients are characterized.

## Abstract

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan-Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan-Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups.