526 research outputs found
Makanin-Razborov diagrams for limit groups
We give a description of , where is a limit group (fully
residually free group). We construct a finite diagram of groups,
Makanin-Razborov diagram, that gives a convinient representation of all such
homomorphisms.Comment: 22 pages, 7 figure
Combination of convergence groups
We state and prove a combination theorem for relatively hyperbolic groups
seen as geometrically finite convergence groups. For that, we explain how to
contruct a boundary for a group that is an acylindrical amalgamation of
relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our
result to Sela's theory on limit groups and prove their relative hyperbolicity.
We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm
A Jorgensen-Thurston theorem for homomorphisms
In this note, we provide a description of the structure of homomorphisms from
a finitely generated group to any torsion-free (3-dimensional) Kleinian group
with uniformly bounded finite covolume. This is analogous to the
Jorgensen-Thurston Theorem in hyperbolic geometry.Comment: 16 pages, 4 figure
A characterization of higher rank symmetric spaces via bounded cohomology
Let be complete nonpositively curved Riemannian manifold of finite volume
whose fundamental group does not contain a finite index subgroup which
is a product of infinite groups. We show that the universal cover is
a higher rank symmetric space iff is injective (and
otherwise the kernel is infinite-dimensional). This is the converse of a
theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem,
as well as a new construction of quasi-homomorphisms on groups that act on
CAT(0) spaces and contain rank 1 elements
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