A compact (orientable or non-orientable) surface of genus g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g. The present paper concerns non-orientable extremal surfaces of genus 5. We represent the surfaces as side-pairing patterns of a hyperbolic regular 24-gon, that is, a generic fundamental region of an NEC group uniformizing...

The moduli space QMg of genus g quadratic differentials has a natural action of G = GL2 (R)/ 〈± ( 1 0 0 1 )〉 and the Veech group PSL(X, q) is the stabilizer of (X, q) ∈ QMg in G. We describe a new algorithm for finding elements of PSL(X, q) which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of gen...

We give a new construction of strongly aperiodic set of tiles in H, exhibiting a kind of hierarchical structure, simplifying the central framework of Margenstern’s proof that the Domino problem is undecidable in the hyperbolic plane [13]. Ludwig Danzer once asked whether, in the hyperbolic plane, where there are no similarities, there could be any notion of hierarchical tiling—an idea which pla...

Consider a group G acting nicely on a simply-connected simplicial complex X . Numerous classical methods exist for using this group action to produce a presentation for G. For the case that X/G is 2-connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbi...

We present an algorithm that can create patterns that are locally fractal in nature, but repeat in two independent directions in the Euclidean plane — in other word “wallpaper patterns”. The goal of the algorithm is to randomly place progressively smaller copies of a basic sub-pattern or motif within a fundamental region for one of the 17 wallpaper groups. This is done in such a way as to compl...

In recently published work Maskit constructs a fundamental domain Dg for the Teichmüller modular group of a closed surface S of genus g ≥ 2. Maskit’s technique is to demand that a certain set of 2g non-dividing geodesics C2g on S satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in C2g must satisfy. Maskit shows that this set of ...

2d gravity stress tensor and the problem of the calculation of the multi-loop amplitudes in the string theory Abstract It is shown that the vacuum value of the 2d gravity stress tensor in the free field theory is singular in the fundamental region on the complex plane where the genus-n > 1 Riemann surface are mapped. Because of the above singularity, one can not construct modular invariant mult...

For SU(2) gauge theory on the three-sphere we implement the influence of the boundary of the fundamental domain, and in particular the θ-dependence, on a subspace of low-energy modes of the gauge field. We construct a basis of functions that respect these boundary conditions and use these in a variational approximation of the spectrum of the lowest order effective hamiltonian.

This paper contains some tentative steps towards describing the structure of non-discrete subgroups of SL(2; R). The main idea is that if a one-parameter family of groups G z varies analytically with the parameter z , then, using analytic continuation, certain results about discrete groups can be analytically continued to those groups in the family that are not discrete. The paper concentrates ...

The unique physical characteristics of coherence, intensity, and monochromatically offered by laser instrumentation has placed renewed importance on studies in photochemistry and photobiology. Sufficient research experience has now been accumulated to demonstrate the potential usefulness of laser energy to fundamental cell biology, and to the diagnosis and treatment of certain pathological cond...