1,133 research outputs found
A generating function of the number of homomorphisms from a surface group into a finite group
A generating function of the number of homomorphisms from the fundamental
group of a compact oriented or non-orientable surface without boundary into a
finite group is obtained in terms of an integral over a real group algebra. We
calculate the number of homomorphisms using the decomposition of the group
algebra into irreducible factors. This gives a new proof of the classical
formulas of Frobenius, Schur, and Mednykh.Comment: 12 pages, 1 figure. Prepared in AMS-LaTe
Entropy of random coverings and 4D quantum gravity
We discuss the counting of minimal geodesic ball coverings of -dimensional
riemannian manifolds of bounded geometry, fixed Euler characteristic and
Reidemeister torsion in a given representation of the fundamental group. This
counting bears relevance to the analysis of the continuum limit of discrete
models of quantum gravity. We establish the conditions under which the number
of coverings grows exponentially with the volume, thus allowing for the search
of a continuum limit of the corresponding discretized models. The resulting
entropy estimates depend on representations of the fundamental group of the
manifold through the corresponding Reidemeister torsion. We discuss the sum
over inequivalent representations both in the two-dimensional and in the
four-dimensional case. Explicit entropy functions as well as significant bounds
on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure
On the number of optimal surfaces
Let X be a closed oriented Riemann surface of genus > 1 of constant negative
curvature -1. A surface containing a disk of maximal radius is an optimal
surface. This paper gives exact formulae for the number of optimal surfaces of
genus > 3 up to orientation-preserving isometry. We show that the automorphism
group of such a surface is always cyclic of order 1,2,3 or 6. We also describe
a combinatorial structure of nonorientable hyperbolic optimal surfaces.Comment: This is the version published by Geometry & Topology Monographs on 29
April 200
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