3,615 research outputs found
Uncountably many quasi-isometry classes of groups of type
Previously one of the authors constructed uncountable families of groups of
type and of -dimensional Poincar\'e duality groups for each .
We strengthen these results by showing that these groups comprise uncountably
many quasi-isometry classes. We deduce that for each there are
uncountably many quasi-isometry classes of acyclic -manifolds admitting free
cocompact properly discontinuous discrete group actions.Comment: Version 2: minor corrections made, theorems now numbered by sectio
Schnyder woods for higher genus triangulated surfaces, with applications to encoding
Schnyder woods are a well-known combinatorial structure for plane
triangulations, which yields a decomposition into 3 spanning trees. We extend
here definitions and algorithms for Schnyder woods to closed orientable
surfaces of arbitrary genus. In particular, we describe a method to traverse a
triangulation of genus and compute a so-called -Schnyder wood on the
way. As an application, we give a procedure to encode a triangulation of genus
and vertices in bits. This matches the worst-case
encoding rate of Edgebreaker in positive genus. All the algorithms presented
here have execution time , hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational
Geometr
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