On Minimal Triangulations of Products of Convex Polygons

Abstract

We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321-339, 2005).

The simplicial volume of closed manifolds covered by ℍ2× ℍ2

“…Furthermore, when M and N are hyperbolic surfaces, the upper bound of 6 • M • N was improved to 3, 25 • M • N by Bowen et al in [Bo&al04] by exhibiting explicit triangulations of products of polygons. The authors also give lower bounds for the minimal number of simplices in such triangulations, which we improve in [Bu07] using the explicit cocycle Θ representing the volume form ω H 2 ×H 2 . Since those triangulations produce fundamental cycles for the fundamental class [M × N ] which in view of Corollary 3 have strictly greater ℓ 1 -norm than M × N , this indicates the existence of triangulations of products of hyperbolic surfaces not arising from triangulations of fundamental domains of the form of a product of polygons.…”

Erratum to: “The Gromov norm of the product of two surfaces” [Topology 44 (2005) 321–339]

“…As an update, in a more recent paper [2] the same author has improved our lower bound for the polytopal Gromov norm of the product of two polygons. We proved P(m, n) ≥ 2mn −O(m+n) and she gets P(m, n) ≥ 3.125mn −5(m+n)+6.…”