1,184 research outputs found
Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes
A quadrangulation is a graph embedded on the sphere such that each face is
bounded by a walk of length 4, parallel edges allowed. All quadrangulations can
be generated by a sequence of graph operations called vertex splitting,
starting from the path P_2 of length 2. We define the degree D of a splitting S
and consider restricted splittings S_{i,j} with i <= D <= j. It is known that
S_{2,3} generate all simple quadrangulations.
Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}.
First we show that the splittings S_{1,2} are exactly the monotone ones in the
sense that the resulting graph contains the original as a subgraph. Then we
show that they define a set of nontrivial ancestors beyond P_2 and each
quadrangulation has a unique ancestor.
Our results have a direct geometric interpretation in the context of
mechanical equilibria of convex bodies. The topology of the equilibria
corresponds to a 2-coloured quadrangulation with independent set sizes s, u.
The numbers s, u identify the primary equilibrium class associated with the
body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate
all primary classes from a finite set of ancestors which is closely related to
their geometric results.
If, beyond s and u, the full topology of the quadrangulation is considered,
we arrive at the more refined secondary equilibrium classes. As Domokos,
L\'angi and Szab\'o showed recently, one can create the geometric counterparts
of unrestricted splittings to generate all secondary classes. Our results show
that S_{1,2} can only generate a limited range of secondary classes from the
same ancestor. The geometric interpretation of the additional ancestors defined
by monotone splittings shows that minimal polyhedra play a key role in this
process. We also present computational results on the number of secondary
classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table
Line Patterns in Free Groups
We study line patterns in a free group by considering the topology of the
decomposition space, a quotient of the boundary at infinity of the free group
related to the line pattern. We show that the group of quasi-isometries
preserving a line pattern in a free group acts by isometries on a related space
if and only if there are no cut pairs in the decomposition space.Comment: 35 pages, 22 figures, PDFLatex; v2. finite index requires extra
hypothesis; v3. 37 pages, 24 figures: updated references and add example in
Section 6.3 of a rigid pattern for which the free group is not finite index
in the group of pattern preserving quasi-isometries; v4. 40 pages, 26
figures: improved exposition and add example in Section 6.4 of a rigid
pattern whose cube complex is not a tre
Mutations and short geodesics in hyperbolic 3-manifolds
In this paper, we explicitly construct large classes of incommensurable
hyperbolic knot complements with the same volume and the same initial (complex)
length spectrum. Furthermore, we show that these knot complements are the only
knot complements in their respective commensurabiltiy classes by analyzing
their cusp shapes.
The knot complements in each class differ by a topological cut-and-paste
operation known as mutation. Ruberman has shown that mutations of hyperelliptic
surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide
geometric and topological conditions under which such mutations also preserve
the initial (complex) length spectrum. This work requires us to analyze when
least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.Comment: This is the final (accepted) version of this pape
Some open questions on anti-de Sitter geometry
We present a list of open questions on various aspects of AdS geometry, that
is, the geometry of Lorentz spaces of constant curvature -1. When possible we
point out relations with homogeneous spaces and discrete subgroups of Lie
groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open
questions. 19 page
Countable groups are mapping class groups of hyperbolic 3-manifolds
We prove that for every countable group G there exists a hyperbolic
3-manifold M such that the isometry group of M, the mapping class group of M,
and the outer automorphism group of the fundamental group of M are isomorphic
to G.Comment: 15 pages, 6 figure
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