Integer hulls of linear polyhedra and scl in families
Danny Calegari, Alden Walker

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.
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