An isometric action of the Cremona group on an infinite dimensional hyperbolic space

2021 ◽  
pp. 11-24
Keyword(s):  
Hyperbolic Space ◽  
Isometric Action ◽  
Cremona Group ◽  
Infinite Dimensional

scholarly journals Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

2011 ◽  
Vol 148 (1) ◽  
pp. 153-184 ◽  
Author(s):  
Thomas Delzant ◽  
Pierre Py
Keyword(s):  
Hyperbolic Space ◽  
Discrete Subgroup ◽  
Hyperbolic Spaces ◽  
Classical Theorem ◽  
Isometric Action ◽  
Cremona Group ◽  
The Real ◽  
Infinite Dimensional ◽  
Kähler Groups ◽  
Real Hyperbolic Space

AbstractGeneralizing a classical theorem of Carlson and Toledo, we prove thatanyZariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

The Bishop–Jones relation and Hausdorff geometry of convex-cobounded limit sets in infinite-dimensional hyperbolic space

2016 ◽  
Vol 16 (05) ◽  
pp. 1650018 ◽  
Author(s):  
Tushar Das ◽  
Bernd O. Stratmann ◽  
Mariusz Urbański
Keyword(s):  
Hyperbolic Space ◽  
Limit Sets ◽  
Multiplicative Constant ◽  
Infinite Dimensional ◽  
Mass Redistribution ◽  
Isometric Actions

We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

scholarly journals Fixed-point theorems for groups acting on non-positively curved manifolds

2019 ◽  
pp. 1-19
Author(s):  
Omer Lavy
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Isometric Action ◽  
Infinite Dimensional ◽  
Curved Manifolds ◽  
Isometric Actions ◽  
Finite Quotient ◽  
Positively Curved

We study isometric actions of Steinberg groups on Hadamard manifolds. We prove some rigidity properties related to these actions. In particular, we show that every isometric action of [Formula: see text] on Hadamard manifold when [Formula: see text] factors through a finite quotient. We further study actions on infinite-dimensional manifolds and prove a fixed-point theorem related to such actions.

scholarly journals Hyperbolic and cubical rigidities of Thompson’s group V

2019 ◽  
Vol 22 (2) ◽  
pp. 313-345 ◽  
Author(s):  
Anthony Genevois
Keyword(s):  
General Criterion ◽  
Isometric Action ◽  
Group V ◽  
Cube Complex ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Fixed Point Properties ◽  
Thompson’S Group ◽  
Cube Complexes ◽  
Finitely Presented

Abstract In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group V is hyperbolically elementary, and we deduce that it satisfies Property {({\rm FW}_{\infty})} , i.e., every isometric action of V on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.

scholarly journals The horofunction boundary of the infinite dimensional hyperbolic space

2019 ◽  
Vol 207 (1) ◽  
pp. 255-263
Author(s):  
Floris Claassens
Keyword(s):  
Hyperbolic Space ◽  
Infinite Dimensional ◽  
Real Hyperbolic Space

AbstractIn this paper we give a complete description of the horofunction boundary of the infinite dimensional real hyperbolic space, and characterise its Busemann points.

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