Generalized representation stability for disks in a strip and no-k-equal spaces

TL;DR

This paper investigates the homology stability of configuration spaces of disks in a strip and points with no k-equal restrictions, proving generalized stability in specific cases and suggesting broader patterns.

Contribution

It establishes generalized representation stability for disks in a width-2 strip and for no-k-equal point configurations, advancing understanding of homology growth patterns.

Findings

Proves stability for width-2 strip disks

Demonstrates stability for no-k-equal point spaces

Shows exponential growth in homology rank

Abstract

For fixed j and w, we study the j-th homology of the configuration space of n labeled disks of width 1 in an infinite strip of width w. As n grows, the homology groups grow exponentially in rank, suggesting a generalized representation stability as defined by Church--Ellenberg--Farb and Ramos. We prove this generalized representation stability for the strip of width 2, leaving open the case of w > 2. We also prove it for the configuration space of n labeled points in the line, of which no k are equal.