This work was partially supported by The Geometry Center of the University of Minnesota and the University of Massachusetts at Amherst. The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vector field. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally mini...

Let X be a finite aspherical CW-complex whose fundamental group π1(X) possesses a subnormal series π1(X) ⊲ Gm ⊲ . . . ⊲ G0 with a non-trivial elementary amenable group G0. We investigate the L2-invariants of the universal covering of such a CW-complex X. We show that the NovikovShubin invariants αn(X̃) are positive. We further prove that the L2-torsion ρ(2)(X̃) vanishes if π1(X) has semi-integral...

Kwak, J.H. and J. Lee, Isomorphism classes of cycle permutation graphs, Discrete Mathematics 105 (1992) 131-142. In this paper, we construct a cycle permutation graph as a covering graph over the dumbbell graph, and give a new characterization of when two given cycle permutation graphs are isomorphic by a positive or a negative natural isomorphism. Also, we count the isomorphism classes of cycl...

Berestovskii and Plaut introduced the concept of a coverable space [1] when developing their theory of generalized universal covering maps for uniform spaces. If a space is coverable and chain connected then it has a generalized universal covering map. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a uniformly joinable space [2] when developing a theory of generalized uniform cover...

If G is a semidirect product N oH with N finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications to 3-manifolds; for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a finite cyclic cover of M whose fundamental group surjects t...

The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every...

The purpose of this article is to clarify the relationship between the algebraic notion of quandle covering introduced by M. Eisermann and the categorical notion of covering arising from Galois theory. A crucial role is played by the adjunction between the variety of quandles and its subvariety of trivial quandles.

In a beautiful paper, Sleator, Tarjan and Thurston solved the problem of maximum rotation distance of two binary trees. Equivalently they solved the problem of rotation distance of triangulations on the disk. We extend their results to rotation distance of triangulations of other planar surfaces. We give upper and lower bounds for this problem. Equivalently, by duality, one can interpret our re...

We begin with some covering theory for complexes of groups. For example, we show that a covering of developable complexes of groups induces a monomorphism of fundamental groups, and an equivariant isometry of universal covers. As an application, we then investigate the asymptotics of the number of “overlattices” of a cocompact lattice Γ in Aut(K), where K is a locally finite polyhedral complex....