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RESEARCH BLOG 9/21/04 BLACK OAK ARKANSAS CONJECTURE
 

Summary: RESEARCH BLOG 9/21/04
BLACK OAK ARKANSAS CONJECTURE
Richard Kent pointed out that there is a much simpler way to see
that the mapping class group of the n-punctured sphere does not have
property T than what I was mentioning in blog 9/20/04. Namely,
there is a finite index subgroup mapping onto Z (actually, one can get
a map onto Z Z, as Kent points out). Take the pure mapping class
group (which doesn't permute punctures), which is of finite index in
the mapping class group, and erase all but 4 points. This gives a map
onto the 4 punctured mapping class group, which is virtually free.
Today, Pete Storm posted a couple of papers at the ArXiv, one of
which gives a solution to a conjecture of Bonahon. This conjecture
has also been formulated by Canary, Taylor and Minsky as the Black
Oak Arkansas conjecture [3]. Apparently, Black Oak Arkansas is a
band that Dick Canary was listening to when he thought of these con-
jectures (see http://www.blackoakarkansas.bigstep.com/). The conjec-
tures state (roughly) that if one has a family of homeomorphic geomet-
rically finite hyperbolic 3-manifolds N, then the volume of the convex
cores C(N) are minimized by 1
2 the Gromov norm of the double along

  

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

 

Collections: Mathematics