78 research outputs found
The genus of curve, pants and flip graphs
This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.Comment: 26 pages, 9 figure
Asymptotically rigid mapping class groups and Thompson's groups
We consider Thompson's groups from the perspective of mapping class groups of
surfaces of infinite type. This point of view leads us to the braided Thompson
groups, which are extensions of Thompson's groups by infinite (spherical) braid
groups. We will outline the main features of these groups and some applications
to the quantization of Teichm\"uller spaces. The chapter provides an
introduction to the subject with an emphasis on some of the authors results.Comment: survey 77
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
On the relation between quantum Liouville theory and the quantized Teichm"uller spaces
We review both the construction of conformal blocks in quantum Liouville
theory and the quantization of Teichm\"uller spaces as developed by Kashaev,
Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert
space acted on by a representation of the mapping class group. According to a
conjecture of H. Verlinde, the two are equivalent. We describe some key steps
in the verification of this conjecture.Comment: Contribution to the proceedings of the 6th International Conference
on CFTs and Integrable Models, Chernogolovka, Russia, September 2002; v2:
Typos corrected, typographical change
The augmented marking complex of a surface
We build an augmentation of the Masur-Minsky marking complex by
Groves-Manning combinatorial horoballs to obtain a graph we call the augmented
marking complex, . Adapting work of Masur-Minsky, we prove
that is quasiisometric to Teichm\"uller space with the
Teichm\"uller metric. A similar construction was independently discovered by
Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy
machinery to to build flexible families of uniform
quasigeodesics in Teichm\"uller space. As an application, we give a new proof
of Rafi's distance formula for the Teichm\"uller metric.Comment: 30 pages; significantly rewritten to strengthen main construction
- …