TY - JOUR
AU - Huszár, Kristóf
PY - 2022/02/02
Y2 - 2022/11/29
TI - On the pathwidth of hyperbolic 3-manifolds
JF - Computing in Geometry and Topology
JA - CompGeomTop
VL - 1
IS - 1
SE - Original Research Articles
DO - 10.57717/cgt.v1i1.4
UR - https://cgt-journal.org/index.php/cgt/article/view/4
SP - 1:1-1:19
AB - <p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been subject of investigations for half a century.</p><p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">Motivated by the algorithmic study of 3-manifolds, Maria and Purcell have recently shown that every closed hyperbolic 3-manifold M with volume vol(M) admits a triangulation with dual graph of treewidth at most C vol(M), for some universal constant C.</p><p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">Here we improve on this result by showing that the volume provides a linear upper bound even on the pathwidth of the dual graph of some triangulation, which can potentially be much larger than the treewidth. Our proof relies on a synthesis of tools from 3-manifold theory: generalized Heegaard splittings, amalgamations, and the thick-thin decomposition of hyperbolic 3-manifolds. We provide an illustrated exposition of this toolbox and also discuss the algorithmic consequences of the result.</p>
ER -