Incompressible fillings of manifolds

TL;DR

This paper investigates the boundaries of certain compactifications of locally symmetric spaces, demonstrating their incompressibility and using small singular models to obstruct symmetries and simplify existing theorems.

Contribution

It introduces the concept of small singular models for boundaries, showing their role in obstructing compressions and symmetries, and applies this to simplify proofs of known theorems.

Findings

Boundaries of Borel-Serre compactifications are incompressible.

Small singular models obstruct $S^1$-actions and homotopically trivial $ ext{Z}/p$-actions.

Simplification of proofs for the minimal orbifold theorem and Royden's theorem.

Abstract

We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$-actions (and more generally homotopically trivial $\mathbb Z/p$-actions) on interiors of aspherical fillings. We use this to bound the symmetry of complete Riemannian metrics on such interiors in terms of the fundamental group. We also use small singular models to simplify the proofs of some already known theorems about moduli spaces (the minimal orbifold theorem and a topological analogue of Royden's theorem).