The infinite dimensional hyperbolic space ℍ∞ does not have property A

2010 ◽  
Vol 31 (4) ◽  
pp. 491-496
Author(s):  
Zhaobo Huang
Keyword(s):  
Hyperbolic Space ◽  
Property A ◽  
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 THE QUANTUM H3 INTEGRABLE SYSTEM

2010 ◽  
Vol 25 (30) ◽  
pp. 5567-5594 ◽  
Author(s):  
MARCOS A. G. GARCÍA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Integrable System ◽  
Differential Operators ◽  
Three Dimensional ◽  
Integrable Model ◽  
Dimensional System ◽  
Property A ◽  
Algebraic Form ◽  
Infinite Dimensional ◽  
Polynomial Coefficients ◽  
Finite Dimensional

The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.

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 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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