The Strong Symmetric Genus Spectrum of Abelian Groups

The strong symmetric genus of a group G is the minimum genus of any compact surface on which G acts faithfully while preserving orientation. We investigate the set of positive integers which occur as the strong symmetric genus of a finite abelian group. This is called the strong symmetric genus spectrum. We prove that there are an infinite number of gaps in the strong symmetric genus spectrum of finite abelian groups. We also determine an upper bound for the size of a finite abelian group that can act faithfully on a surface of a particular genus and then find the genus of abelian groups in particular families. These formulas produce a lower bound for the density of the strong symmetric genus spectrum

Orbifold genera, product formulas and power operations

We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H-infinity-map into the Morava-Lubin-Tate theory E_h, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus. It depends only on h and not on the genus.Comment: 36 pages, PhD thesis, revised versio

Weakly commensurable groups, with applications to differential geometry

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note: substantial text overlap with arXiv:1212.121

Finite length spectra of random surfaces and their dependence on genus

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we study the probability distribution of the systole of random surfaces in a hyperbolic and a more general Riemannian setting. In the hyperbolic setting we are able to determine the limit of the probability distribution for the number of triangles tending to infinity and in the Riemannian setting we derive bounds.Comment: 30 pages, 6 figure

On K(1)-local SU-bordism

This paper works towards a K(1)-local multiplicative splitting of SU-bordism

Simple closed geodesics and the study of Teichm\"uller spaces

The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichm\"uller spaces.Comment: to appear in Handbook of Teichm\"uller theory, vol II