Stability phenomena in the homology of tree braid groups

TL;DR

This paper investigates the homology groups of tree braid groups, showing their ranks follow a polynomial pattern and providing explicit formulas, thus revealing stability phenomena in their algebraic structure.

Contribution

It introduces a polynomial description of homology ranks for tree braid groups and constructs a graded module structure, advancing understanding of their algebraic properties.

Findings

Homology ranks are polynomial in n for all n.

The homology groups form a finitely generated graded module.

Decomposition into shifts of squarefree monomial ideals.

Abstract

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := \pi_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and construct this polynomial explicitly in terms of invariants of the tree $G$. To accomplish this we prove that the group $\bigoplus_n H_i(B_nG)$ can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Following this, we spend time considering how our methods might be generalized to braid groups of arbitrary graphs, and make various conjectures in this direction.