Critical levels and Jacobi fields in a complex of cycles

TL;DR

This paper introduces a finite-dimensional structure for the space of tight geodesic segments in a complex of cycles, using discrete Jacobi fields, without relying on hyperbolicity, and offers a Morse-theoretic approach.

Contribution

It constructs explicit discrete Jacobi fields to fully describe the space of tight geodesics in a complex of cycles, establishing a uniform bound on its dimension.

Findings

Finite, uniformly bounded dimension of geodesic space

Explicit construction of discrete Jacobi fields

Morse-theoretic approach independent of hyperbolicity

Abstract

In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the entire space of tight geodesics. Jacobi fields measure the extent to which geodesic stability breaks down. Unlike most finiteness properties of curve complexes, the arguments presented here do not rely on hyperbolicity, but rather on structures similar to Morse theory.