328 research outputs found
Balanced walls for random groups
We study a random group G in the Gromov density model and its Cayley complex
X. For density < 5/24 we define walls in X that give rise to a nontrivial
action of G on a CAT(0) cube complex. This extends a result of Ollivier and
Wise, whose walls could be used only for density < 1/5. The strategy employed
might be potentially extended in future to all densities < 1/4.Comment: 18 pages, 2 figures. v2: Minor improvements, final versio
On the Minimum Ropelength of Knots and Links
The ropelength of a knot is the quotient of its length and its thickness, the
radius of the largest embedded normal tube around the knot. We prove existence
and regularity for ropelength minimizers in any knot or link type; these are
curves, but need not be smoother. We improve the lower bound for the
ropelength of a nontrivial knot, and establish new ropelength bounds for small
knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and
corrections; new section on asymptotic growth of ropelength; several new
reference
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