Let
be a closed oriented surface of negative Euler characteristic and
a complete contractible Riemannian manifold. A Fuchsian representation
strictly dominates a
representation
if there
exists a
–equivariant
map from
to
that is
–Lipschitz
for some
.
In a previous paper by Deroin and Tholozan, the authors construct a map
from the Teichmüller
space
of the surface
to itself and prove that,
when
has sectional
curvature at most
, the
image of
lies (almost
always) in the domain
of Fuchsian representations strictly dominating
. Here we prove
that
is a
homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of
representations
from
to
with
Fuchsian strictly
dominating
.
In particular, we obtain that its connected components are classified by the Euler class
of
. The
link with anti-de Sitter geometry comes from a theorem of Kassel, stating that those
pairs parametrize deformation spaces of anti-de Sitter structures on closed
–manifolds.
Keywords
anti-de Sitter, representations of surface groups,
Teichmüller, harmonic maps, deformation space
