18,256 research outputs found
Nucleation-free rigidity
When all non-edge distances of a graph realized in as a {\em
bar-and-joint framework} are generically {\em implied} by the bar (edge)
lengths, the graph is said to be {\em rigid} in . For ,
characterizing rigid graphs, determining implied non-edges and {\em dependent}
edge sets remains an elusive, long-standing open problem.
One obstacle is to determine when implied non-edges can exist without
non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal
with them.
In this paper, we give general inductive construction schemes and proof
techniques to generate {\em nucleation-free graphs} (i.e., graphs without any
nucleation) with implied non-edges. As a consequence, we obtain (a) dependent
graphs in that have no nucleation; and (b) nucleation-free {\em
rigidity circuits}, i.e., minimally dependent edge sets in . It
additionally follows that true rigidity is strictly stronger than a tractable
approximation to rigidity given by Sitharam and Zhou
\cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial
characterization.
As an independently interesting byproduct, we obtain a new inductive
construction for independent graphs in . Currently, very few such inductive
constructions are known, in contrast to
On the quasi-isometric rigidity of graphs of surface groups
Let be a word hyperbolic group with a cyclic JSJ decomposition that
has only rigid vertex groups, which are all fundamental groups of closed
surface groups. We show that any group quasi-isometric to is
abstractly commensurable with .Comment: 54 pages, 10 figures, comments welcom
Counting generalized Jenkins-Strebel differentials
We study the combinatorial geometry of "lattice" Jenkins--Strebel
differentials with simple zeroes and simple poles on and of the
corresponding counting functions. Developing the results of M. Kontsevich we
evaluate the leading term of the symmetric polynomial counting the number of
such "lattice" Jenkins-Strebel differentials having all zeroes on a single
singular layer. This allows us to express the number of general "lattice"
Jenkins-Strebel differentials as an appropriate weighted sum over decorated
trees.
The problem of counting Jenkins-Strebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for the
volume of any stratum of meromorphic quadratic differentials recently obtained
by the authors leads to an interesting combinatorial identity for our sums over
trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with
arXiv:1212.166
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