Representation Stability for Marked Graph Complexes

TL;DR

This paper establishes a precise representation stability result for graph complexes with a marked vertex, showing that certain homology classes persist in a stable manner, extending known results to higher genus cases.

Contribution

It provides a sharp stability bound for marked graph complexes and demonstrates the non-triviality of associated graph homology classes, generalizing previous stability results.

Findings

Proves a sharp representation stability bound for marked graph complexes.

Shows that chains realizing this bound lead to non-trivial graph homology classes.

Generalizes stability results to higher genus configurations.

Abstract

We prove a sharp representation stability result for graph complexes with a distinguished vertex, and prove that the chains realizing this sharp bound pass to non-trivial families of graph homology classes. This result may be interpreted as a higher genus generalization of Hersh and Reiner's stability bound for configuration spaces of points in odd dimensional Euclidean space.