Edge (group)
In mathematics , the edge of a group is an important tool in representation theory . Representations of groups can often be examined by means of their marginal images.
The general definition given here generalizes the Furstenberg-Poisson boundary of locally compact groups and also the boundary at infinity of hyperbolic fundamental groups.
definition
Be a topological group . A G-space with an invariant Lebesgue measure is called the edge of G if it is an indirect G-space and the projection onto the first or second factor is relatively metric ergodic .
existence
Every locally compact group that satisfies the second countability axiom has a boundary. It corresponds to the Furstenberg-Poisson margin of a symmetrical, spanning dimension.
Examples
- The Furstenberg edge of a semi- simple Lie group is for a Borel subgroup . Each edge of is an equivariate picture of .
- If the boundary is a locally compact group , then it is also a boundary for every lattice .
- Let be a compact Riemann manifold of negative section curvature and its universal superposition . Then the margin at infinity is a margin of with a Patterson-Sullivan measure .
Edge illustration
definition
Let be a countable group with a border . Then there is each representation
with an unlimited and Zariski - dense image in the general linear group over a local body a clear, measurable , - equivariant mapping
into the flag manifold .
Examples
- Anosov representations have a continuous edge mapping.
- The edge mapping of the only irreducible representation is an embedding generated by a hyperconvex curve , the generating first component of which is the Veronese embedding .
literature
- U. Bader, A. Furman: Boundaries, rigidity of representations, and Lyapunov exponents , Proceedings of ICM 2014, Invited Lectures, (2014), 71-96.
- U. Bader, A. Furman: Boundaries, Weyl groups, and Superrigidity , Electron. Res. Announc. Math. Sci., Vol 19 (2012), 41-48.
- U. Bader, B. Duchesne, J. Lcureux (2014). Furstenberg Maps for CAT (0) Targets of Finite Telescopic Dimension.