Nakamura [N] introduced the G-Hilbert scheme G-Hilb C3 for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-HilbC3. This note calculates A-Hilb C3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilate...

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full G-shift for a finitely-generated torsion-free nilpotent group G. Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape

Let ω be a point in the upper half plane, and let Γ be a discrete, finite covolume subgroup of PSL2(R). We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice Γω, intersected with increasingly large balls centered at ω. We prove this conjecture for Γ = PSL2(Z) and ω an elliptic point.

By “Euclidean space” of dimension n we mean a real vector space of dimension n, equipped with a positive-definite inner product 〈·, ·〉. We usually call such a space “Rn” even when there is no distinguished choice of coordinates. A lattice in Rn is a discrete co-compact subgroup L ⊂ Rn, that is, a discrete subgroup such that the quotient Rn/L is compact (and thus necessarily homeomorphic with th...

Let [Formula: see text] be the simple group text], where is a prime number. For any subgroup of we compute Möbius function in lattice text]. To this aim, describe intersections maximal subgroups We point out some connections with other combinatorial objects, and, context, reduced Euler characteristic order complex subposet text]-subgroups for and power

In Quantum Computation and Lattice Problems [11] Oded Regev presented the first known connection between lattices and quantum computation, in the form of a quantum reduction from the poly(n)-unique shortest vector problem to the dihedral hidden subgroup problem by sampling cosets. This article contains a summary of Regev’s result.

This talk connects simple lattice path enumeration with a subgroup of the Riordan group, ordered trees, and the Faber polynomials from classical complex analysis. The main tools employed are matrix multiplication, generating functions and a few definitions from group theory and complex functions.

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and thei...

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, have positive first Betti number, a homomorphisms onto non-Abelian free group. For we determine minimal neat subgroup. Finally, answer question raised by Stover describe an infinite tower ball quotients all with single cusp.