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RESEARCH BLOG 7/19/04 A paper is published by Ruth Kellerhals, which proves that the mini-
 

Summary: RESEARCH BLOG 7/19/04
A paper is published by Ruth Kellerhals, which proves that the mini-
mal volume hyperbolic 4-orbifold is that Q = H4
/ , where is the co-
compact arithmetic Coxeter group ---- ---- ---- --5---- . The
volume of Q equals 2
/10, 800 . This corroborates the phenomenolog-
ical observation that minimal volume hyperbolic objects tend to be
arithmetic. Since in even dimensions, the volume of a hyperbolic orb-
ifold is proportional to its Euler characteristic, it is natural to wonder
whether this orbifold has minimal Euler characteristic among aspherical
"good" 4-orbifolds (ones with an aspherical manifold universal cover)
with positive Euler characteristic (it is natural to conjecture that the
Euler characteristic of such orbifolds is always non-negative ). It is
also natural to wonder whether the set of Euler characteristics of as-
pherical good 2n-orbifolds is well-ordered. Certainly this is true in two
dimensions, and seems to be consistent with what is known in higher
dimensions, e.g. the examples of Gromov and Thurston. These ex-
amples are pinched negatively curved 4-orbifolds, which are obtained
by introducing an order n cone singularity along a geodesic surface in

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics