Effective mapping class group dynamics, I: Counting lattice points in Teichmüller space

Lipschitz class, Narrow class, and counting lattice points

A well known principle says that the number of lattice points in a bounded subsets S of Euclidean space is about the ratio of the volume and the lattice determinant, subject to some relatively mild conditions on S. In the literature one finds two different types of such conditions; one asserts the Lipschitz parameterizability of the boundary ∂S, and the other one is based on intersection proper...

Dynamics over Teichmüller Space

Counting Lattice Points in the Moduli Space of Curves

where the indexing set Fatg,n is the space of labeled fatgraphs of genus g and n boundary components. See Section 2 for definitions of a fatgraph Γ, its automorphism group AutΓ and the cell decomposition (1) realised as the space of labeled fatgraphs with metrics. Restricting this homeomorphism to a fixed n-tuple of positive numbers (b1, ..., bn) yields a space homeomorphic to Mg,n decomposed i...

Counting Lattice Points in Polyhedra

We present Barvinok’s 1994 and 1999 algorithms for counting lattice points in polyhedra. 1. The 1994 algorithm In [2], Barvinok presents an algorithm that, for a fixed dimension d, calculates the number of integer points in a rational polyhedron. It is shown in [6] and [7] that the question can be reduced to counting the number of integer points in a k-dimensional simplex with integer vertices ...

Universal Counting of Lattice Points in Polytopes

Given a lattice polytope P (with underlying lattice L), the universal counting function UP (L ) = |P ∩ L| is defined on all lattices L containing L. Motivated by questions concerning lattice polytopes and the Ehrhart polynomial, we study the equation UP = UQ. Mathematics Subject Classification: 52B20, 52A27, 11P21 Partially supported by Hungarian Science Foundation Grant T 016391, and by the Fr...