Richard Evan Schwartz

The quasi-isometry classification of rank one lattices

Let X be a symmetric space—other than the hyperbolic plane—of strictly negative sectional curvature. Let G be the isometry group of X. We show that any quasi-isometry between non-uniform lattices in G is equivalent to (the restriction of) a group element of G which commensurates one lattice to the other. This result has the following corollaries:1.Two non-uniform lattices in G are quasi-isometric if and only if they are commensurable.2.Let Γ be a finitely generated group which is quasi-isometric to a non-uniform lattice in G. Then Γ is a finite extension of a non-uniform lattice in G.3.A non-uniform lattice in G is arithmetic if and only if it has infinite index in its quasi-isometry group.

The Pentagram Map

We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagons diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a “pentagram map” on convex polygons of more than five sides, showing that iterated images of anyinitial polygon converge exponentially fast to a point. We conjecture that the asymptotic behavior of this convergence is the same as under a projective transformation. Finally, we show a connection between the pentagram map and a certain flow defined on parametrized curves.

Liouville–Arnold integrability of the pentagram map on closed polygons

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems. Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

Degenerating the complex hyperbolic ideal triangle groups

A basic problem in geometry and representation theory is the deformation problem. Suppose that A0:F~+G1 is a discrete embedding of a finitely generated group F into a Lie group G1. Suppose also that GICG2, where G2 is a larger Lie group. The deformation problem amounts to finding and studying discrete embeddings As: F-~G2 which extend AoLet H 2 be the hyperbolic plane. The complex hyperbolic plane, C H 2, is a complex 2-dimensional negatively curved symmetric space which contains H 2 as a totally real, totally geodesic subspace, and is often considered to be its complexification. The theory of deforming Isom(H2)-representations into Isom(CH2), while quite rich, is still in its infancy. (For a representative sample of such work, see [FZ], [GKL], [GuP], [KR], [To].) The state of affairs is such that one still needs to work out basic examples in detail to gain a foundation for more general considerations. The complex hyperbolic ideal triangle groups are amongst the simplest concrete examples of complex hyperbolic deformations. A complex hyperbolic ideal triangle group is a representation of the form As: F-+Isom(CH2). Here F is the free product Z / 2 . Z / 2 * Z / 2 . The representation As maps the standard generators to order-2 complex reflections, such that the product of any two unequal generators is parabolic. (See w for definitions.) There is a real 1-parameter family {A~ l sER} of nonconjugate complex hyperbolic ideal triangle groups. The representation A0 is the complexification of the familiar real ideal triangle group generated by reflections in the sides of an ideal geodesic triangle in the hyperbolic plane. The other representations are deformations.

Ideal triangle groups, dented tori, and numerical analysis

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is discretely embedded in PU(2, 1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates.

The Poncelet grid

Given a convex polygon P in the projective plane we can form a finite “grid” of points by taking the pairwise intersections of the lines extending the edges of P . When P is a Poncelet polygon we show that this grid is contained in a finite union of ellipses and hyperbolas and derive other related geometric information about the grid.

The Five-Electron Case of Thomson’s Problem

We give a rigorous computer-assisted proof that the triangular bipyramid is the unique configuration of five points on the sphere that globally minimizes the Coulomb (1/r) potential. We also prove the same result for the (1/r 2) potential. The main mathematical contribution of the paper is a fairly efficient energy estimate that works for any number of points and any power-law potential.

The Pentagram Map is Recurrent

The pentagram map is defined on the space of convex n-gons (considered up to projective equivalence) by drawing the diagonals that join second-nearest-neighbors in an n-gon and taking the new n-gon formed by the intersections. We prove that th is map is recurrent; thus, for almost any starting polygon, repeated application of the pentagram map will show a near copy of the starting polygon appear infinitely often under various perspectives.

Obtuse Triangular Billiards I: Near the

Let Sε denote the set of Euclidean triangles whose two small angles are within ε radians of and respectively. In this paper we prove two complementary theorems: (1) For any ε > 0 there exists a triangle in Sε that has no periodic billiard path of combinatorial length less than 1/ε. (2) Every triangle in S 1/400 has a periodic billiard path.

(2,3,6)

We give a rigorous computer-assisted proof that a triangle has a periodic billiard path when all its angles are at most one hundred degrees.