VEECH GROUPS FOR HOLONOMY-FREE TORUS COVERS

For fixed coprime k, l ∈ ℕ and each pair (w, z) ∈ ℂ2we define an infinite cyclic cover Σk,l(w, z) → 𝕋, called a k-l-surface or k-l-cover. We show that [Formula: see text] classifies k-l-covers up to isomorphism away from a rather small set. The diagonal action of SL2(ℤ) on ℂ2descends to [Formula: see text], reflecting the SL2(ℤ)-action on the family of k-l-surfaces equipped with a translation structure. The moduli space of holonomy free k-l-surfaces is a compact SL2(ℤ) invariant subspace [Formula: see text] containing all k-l-surfaces with a lattice stabilizer with respect to the SL2(ℤ) action. We calculate the stabilizer, the Veech group, explicitly and represent k-l-covers branched over two points by a generalized class of staircase surfaces. Finally we study SL2(ℤ)-equivariant translation maps from the Hurwitz space of k-(d - k)-covers to Hurwitz spaces of ℤ/d-covers branched over two points.

FINITE COVERS OF THE INFINITE CYCLIC COVER OF A KNOT

We show that the commutator subgroup G′ of a classical knot group G need not have subgroups of every finite index, but it will if G′ has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a knotted Sn in Sn+2 for all n ≥ 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

Topological Quantum Field Theory and Strong Shift Equivalence

Given a TQFT in dimensiond+ 1; and an infinite cyclic covering of a closed (d+ 1)-dimensional manifoldM, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary ofMhas anS1factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite groupGwhich has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated toGin its unmodified form.

Knots which are branched cyclic covers of only finitely many knots

In [5] we proved two results: theorem 1, which said that if k was a simple (2q – 1)-knot, q 1, then it was equivalent to the m-fold branched cyclic cover of another knot if and only if there existed an isometry u of its Blanchfield pairing 〈,〉, whose mth power was the map induced by a generator t of the group of covering translations associated with the infinite cyclic cover of k; and theorem 2, which showed that if k were the m-fold b.c.c. of two such knots, then these would be equivalent if and only if the corresponding isometries were conjugate by an isometry of 〈,〉. Using this second result, we present two cases where k may only be the m-fold b.c.c. of finitely many knots.