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Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.
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First cohomology of pure mapping class groups of big genus one and zero surfaces
We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to Z factors through a sphere with finitely many punctures. In fact, we get an uncountable family of such maps.
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- Award ID(s):
- 1840190
- PAR ID:
- 10154630
- Date Published:
- Journal Name:
- New York journal of mathematics
- Volume:
- 26
- ISSN:
- 1076-9803
- Page Range / eLocation ID:
- 322-333
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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