53 research outputs found
Topological Invariants of Anosov Representations
We define new topological invariants for Anosov representations and study
them in detail for maximal representations of the fundamental group of a closed
oriented surface into the symplectic group.Comment: 66 pages, several changes, some consequences adde
Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups
The first part of this paper surveys several characterizations of
Teichm\"uller space as a subset of the space of representation of the
fundamental group of a surface into PSL(2,R). Special emphasis is put on
(bounded) cohomological invariants which generalize when PSL(2,R) is replaced
by a Lie group of Hermitian type. The second part discusses underlying
structures of the two families of higher Teichm\"uller spaces, namely the space
of maximal representations for Lie groups of Hermitian type and the space of
Hitchin representations or positive representations for split real simple Lie
groups.Comment: The file uploaded on May 12th was the wrong one and did not contain
the Section 4.6 that was added. This is the version to appear in the Handbook
of Teichm\"uller theor
Limits of geometries
A geometric transition is a continuous path of geometric structures that
changes type, meaning that the model geometry, i.e. the homogeneous space on
which the structures are modeled, abruptly changes. In order to rigorously
study transitions, one must define a notion of geometric limit at the level of
homogeneous spaces, describing the basic process by which one homogeneous
geometry may transform into another. We develop a general framework to describe
transitions in the context that both geometries involved are represented as
sub-geometries of a larger ambient geometry. Specializing to the setting of
real projective geometry, we classify the geometric limits of any sub-geometry
whose structure group is a symmetric subgroup of the projective general linear
group. As an application, we classify all limits of three-dimensional
hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and
Sol geometry among the limits. We prove, however, that the other Thurston
geometries, in particular and
, do not embed in any limit of
hyperbolic geometry in this sense.Comment: 40 pages, 2 figures. new in v2: figure 2 added, minor edits to
Sections 1,2,
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