We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply co...

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert’s third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of pol...

We show that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non-unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentz 3-manifolds with non compact (local) i...

In [MS, Theorem 1], M. P. Murthy–R. G. Swan have shown that a stably free projective module over a two-dimensional affine variety A over an algebraically closed field k is free. (The example of the tangent bundle over the real two sphere shows that the condition that the base field is algebraically closed is necessary.) They showed that every unimodular vector (a, b, c) ∈ Um3(A) could be transf...

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := ( pn : pn(z) = n X k=0 akz , ak ∈ C , |ak| = 1 ) . The class Kn is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (εn) of positive numbers tending to 0, we say that a sequence (Pn) of unimodular polynomials Pn ∈ Kn i...

The paradigmatic result in symplectic toric geometry is the paper of Delzant that classifies compact connected symplectic manifolds with effective completely integrable torus actions, the so called (compact) symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a simple unim...

Let p ∈ [1,∞[ and cp = maxa∈[0,1]((1 − a)ap + a(1 − a)p)1/p. We prove that the known upper bound lindiscp(A) ≤ cp for the Lp linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e., sup A lindiscp(A) = cp. We estimate cp = p p+1 ( 1 p+1 )1/p (1+εp) for some εp ∈ [0, 2−p+2], hence cp = 1− ln p p (1+ o(1)). We also show that an improvement for smaller matrices as in the ...

in this work we deal with the coefficients of a (e it ) 2 , where a is in a class of polynomialshaving unimodular coefficients. we first present a technique that calculates lower bounds for particularautocorrelations and then in a more general case we present an upper bound for their maximal order.