Flexible exponents of non-geometric 3-manifolds

TL;DR

This paper determines the flexible exponent (M) for non-geometric 3-manifolds, extending previous results known for geometric 3-manifolds, and explores bounds on mapping degree in relation to Lipschitz constants.

Contribution

It extends the understanding of flexible exponents (M) to non-geometric 3-manifolds, providing new bounds and insights in quantitative topology.

Findings

(M) is explicitly determined for non-geometric 3-manifolds.

The results generalize previous work on geometric 3-manifolds.

Bounds on mapping degree in terms of Lipschitz constants are established.

Abstract

A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\text{Lip}(f)$. For a closed, orientable, Riemannian manifold $M$, the flexible exponent $\alpha(M)$ is the infimum of $\alpha\geqslant 0$ such that $|\text{deg}(f)|\leqslant C\cdot (\text{Lip}(f))^\alpha$ holds for any Lipschitz map $f:M\to M$. For a geometric 3-manifold $M$ in the sense of Thurston, $\alpha(M)$ is determined in \cite{DLWWW}. In this paper, we determine $\alpha(M)$ for non-geometric 3-manifolds.