Brent Everitt

Regular Maps on Non-orientable Surfaces

It is well known that regular maps exist on the projective plane but not on the Klein bottle, nor the non-orientable surface of genus 3. In this paper several infinite families of regular maps are constructed to show that such maps exist on non-orientable surfaces of over 77 per cent of all possible genera.

Alternating quotients of the (3.q.r) triangle groups

A long standing conjecture (attributed to Graham Higman) asserts that each of the triangle groups △(p,q,r)for 1/p+1/q+1/r>1 contains among its homomorphic images all but finitely many of the alternating or symmetric groups. This phenomenon has been termed property H by Mushtaq and Servatius [9]. The work of several authors over the last decade and a half has shown that for any value of q, there are only finitely many r such that△(2,q,r) fails to have property H. In this paper, the techniques used by these authors are generalised to handle the possinblity that p is odd, and as a result, it is shown that for any q≧3, there are only finitely many r such that △(3,q,r)fails to have property H.

The homotopy theory of Khovanov homology

We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology. 57M27; 55P42 Motivation and introduction In order to apply the methods of homotopy theory to Khovanov homology there are several natural approaches. One is to build a space or spectrum whose classical invariants give Khovanov homology, then show its homotopy type is a link invariant, and finally study this space using homotopy theory. Ideally this approach would begin with some interesting geometry and lead naturally to Khovanov homology. One also might hope to construct something more refined than Khovanov homology in this way (see Lipshitz and Sarkar [12] for a combinatorial approach to this). Another approach is to interpret the existing constructions of Khovanov homology in homotopy theoretic terms. By placing the constructions into a homotopy setting one makes Khovanov homology amenable to the methods and techniques of homotopy theory. In this paper our interest is with the second of these approaches. Our aim is to show that Khovanov homology can be interpreted in a homotopy theoretic way using homotopy limits and to subsequently develop a number of results about the specific type of homotopy limit arising. The latter will provide homotopy tools appropriate for studying Khovanov homology.

A family of conformally asymmetric Riemann surfaces

We give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g a 3 . The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3, r) for certain values of r.

Computational and Geometric Aspects of Modern Algebra: Constructing hyperbolic manifolds

In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups. We determine when the kernel of such a representation is torsion-free and thus the quotient a hyperbolic n-manifold.

Graphs, free groups and the Hanna Neumann conjecture

Abstract A new bound for the rank of the intersection of finitely generated subgroups of a free group is given, formulated in topological terms, and in the spirit of Stallings [J. R. Stallings. Topology of finite graphs. Invent. Math. 71 (1983), 551–565.]. The bound is a contribution to the strengthened Hanna Neumann conjecture.

Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.

The smallest hyperbolic 6-manifolds

By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in Everitt and Fountain [Adv. Math. 223 (2010) 1782–1814]. In this paper, we study their presentations as abstract monoids. Along the way, we also find general presentations for certain join-semilattices (as monoids under join), which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the ‘classical’ algebraic monoids.