scholarly journals Rigidity at infinity for lattices in rank-one Lie groups

2018 ◽  
Vol 12 (01) ◽  
pp. 113-130 ◽  
Author(s):  
Alessio Savini
Keyword(s):  
Lie Groups ◽  
Representation Formula ◽  
Rigidity Theorem ◽  
Uniform Lattice ◽  
Maximal Volume ◽  
Reducible Representation ◽  
Totally Geodesic ◽  
Rank One ◽  
Text Component ◽  
Standard Lattice

Let [Formula: see text] be a non-uniform lattice in [Formula: see text] without torsion and with [Formula: see text]. By following the approach developed in [S. Francaviglia and B. Klaff, Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006) 111–124], we introduce the notion of volume for a representation [Formula: see text] where [Formula: see text]. We use this notion to generalize the Mostow–Prasad rigidity theorem. More precisely, we show that given a sequence of representations [Formula: see text] such that [Formula: see text], then there must exist a sequence of elements [Formula: see text] such that the representations [Formula: see text] converge to a reducible representation [Formula: see text] which preserves a totally geodesic copy of [Formula: see text] and whose [Formula: see text]-component is conjugated to the standard lattice embedding [Formula: see text]. Additionally, we show that the same definitions and results can be adapted when [Formula: see text] is a non-uniform lattice in [Formula: see text] without torsion and for representations [Formula: see text], still maintaining the hypothesis [Formula: see text].

scholarly journals Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension

2015 ◽  
Vol 37 (2) ◽  
pp. 539-563 ◽  
Author(s):  
S. KADYROV ◽  
A. POHL
Keyword(s):  
Hausdorff Dimension ◽  
Homogeneous Spaces ◽  
Upper Semicontinuity ◽  
Metric Entropy ◽  
Real Rank ◽  
Semisimple Lie Group ◽  
Uniform Lattice ◽  
Finite Center ◽  
Rank One ◽  
Set Of Points

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.

scholarly journals Orbit equivalence and rigidity of ergodic actions of Lie groups

1981 ◽  
Vol 1 (2) ◽  
pp. 237-253 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lie Groups ◽  
Homogeneous Spaces ◽  
The Other ◽  
Rigidity Theorem ◽  
Simple Groups ◽  
Orbit Equivalence ◽  
Finite Invariant Measure ◽  
Lattice Subgroups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

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