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 . ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle \mu }$ ${\displaystyle B\times B\to B}$

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 . ${\displaystyle G}$ ${\displaystyle G/P}$ ${\displaystyle P\subset G}$ ${\displaystyle G}$ ${\displaystyle G/P}$
• If the boundary is a locally compact group , then it is also a boundary for every lattice . ${\displaystyle B}$ ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$
• 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 . ${\displaystyle M}$ ${\displaystyle {\widetilde {M}}}$ ${\displaystyle \pi _{1}M}$

Edge illustration

definition

Let be a countable group with a border . Then there is each representation ${\displaystyle \Gamma }$ ${\displaystyle B}$

${\displaystyle \rho \colon \Gamma \to GL(n,k)}$

with an unlimited and Zariski - dense image in the general linear group over a local body a clear, measurable , - equivariant mapping ${\displaystyle k}$ ${\displaystyle \rho }$

${\displaystyle \xi \colon B\to Flag(k^{n+1})}$

into the flag manifold . ${\displaystyle Flag(k^{n+1})}$

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 . ${\displaystyle \rho \colon SL(2,\mathbb {R} )\to GL(n,\mathbb {R} )}$ ${\displaystyle \xi =(\xi _{1},\ldots ,\xi _{n})\colon P^{1}\mathbb {R} \to Flag(\mathbb {R} ^{n+1})}$ ${\displaystyle \xi _{1}\colon P^{1}\mathbb {R} \to P^{n}\mathbb {R} }$

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.