 
Summary: RESEARCH BLOG 7/19/04
A paper is published by Ruth Kellerhals, which proves that the mini
mal volume hyperbolic 4orbifold 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" 4orbifolds (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 nonnegative ). It is
also natural to wonder whether the set of Euler characteristics of as
pherical good 2norbifolds is wellordered. 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 4orbifolds, which are obtained
by introducing an order n cone singularity along a geodesic surface in
