Actions and representations of mapping class groups – ModGroup

We use various methods inspired from quantum topology
in this study.

There were several results obtained so far eg Witten's conjecture for
many hyperbolic manifolds (Marche and coll) and the study
of finite representations (Masbaum and coll, Funar).

We want to understand the quantum repersentations and
the dynamics on moduli spaces.

(Masbaum, A. W. Reid) All finite groups are involved in the Mapping Class Group. Geometry and Topology (to appear)

(Masbaum, P. M. Gilmer) Dimension formulas for some modular representations of the symplectic group in the natural characteristic. Journal of Pure and Applied Algebra (to appear)

(Funar,W. Pitsch) Finite quotients of symplectic groups vs mapping class groups, 33p., arxiv:1103.1855, revision June 2012.

The emergence of topological quantum field theory in mathematics
can be traced to the work of E. Witten at the end of the 80's.
Prior to this M.Atiyah, N.Hitchin and S.Donaldson
had shown that Yang-Mills Theory (the standard model of particles)
was of profound mathematical interest
providing a  powerful tool to analyse the geometry
of four dimensional manifolds.
The success of their approach was breathtaking.
The Witten's results added force to the
notion that one can obtain results in low dimensional topology
by applying techniques from quantum field theory.

Broadly speaking, the underlying  principal evoked in their work
is that topology in dimensions three and four should
be considered as a branch  of physics,
more particularly of quantum field theory.
Following this realisation, a cascade of results were proven
and new directions of research were opened, giving rise to
what one might call the ``quantum world'':
quantum groups, quantum invariants etc.

Our philosophy in this project, is to apply techniques and lessons
of the quantum approach to low dimensional topology
more widely in the study of actions and representations of the
mapping class groups of surfaces.
A fundamental difficulty in the study  of the the mapping class group
is that we know very few linear representations.
The mapping class group does admit natural actions
on spaces of structures related to the surface; for example
the space of laminations  of the surface,
the curve and arc complexes,
and the character variety of the fundamental group.
Our aim is to study these actions together with
the actions on the associated quantifications of the underlying spaces,
 exploiting  the interplay between geometry, topology and dynamics,
leading to  new perspectives in the theory of the mapping class group.