Integer hulls of linear polyhedra and scl in families

TL;DR

This paper demonstrates that the vertices of integer hulls of rational polyhedra in families have quasipolynomial coordinates, and applies this to analyze stable commutator length in surgery families, showing quasi-polynomial behavior.

Contribution

It establishes quasipolynomial bounds on integer hull vertices and connects this to stable commutator length in families of groups, a novel link between polyhedral geometry and group theory.

Findings

Vertices of integer hulls have quasipolynomial coordinates

Stable commutator length in surgery families is a ratio of quasipolynomials

Unit balls in the scl norm quasi-converge in finite dimensional families

Abstract

The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio of quasipolynomials, and that unit balls in the scl norm quasi-converge in finite dimensional surgery families.