Geometry of compactifications of locally symmetric spaces

Lizhen Ji[1]; Robert Macpherson[2]

  • [1] University of Michigan, Department of Mathematics, Ann Arbor MI 48109-1003 (USA)
  • [2] Institute for Advanced Study, School of Mathematics, Princeton NJ 08540 (USA)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 457-559
  • ISSN: 0373-0956

Abstract

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For a locally symmetric space M , we define a compactification M ∪ M ( ∞ ) which we call the “geodesic compactification”. It is constructed by adding limit points in M ( ∞ ) to certain geodesics in M . The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give M ( ∞ ) for locally symmetric spaces. Moreover, M ( ∞ ) has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in the harmonic analysis of the locally symmetric space:1) it is the minimal Martin compactification for negative eigenvalues of the Laplacian, and 2) it can be used to parameterize the eigenfunctions of the Laplacian in continuous spectrum on L 2 .

How to cite

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Ji, Lizhen, and Macpherson, Robert. "Geometry of compactifications of locally symmetric spaces." Annales de l’institut Fourier 52.2 (2002): 457-559. <http://eudml.org/doc/115986>.

@article{Ji2002,
abstract = {For a locally symmetric space $M$, we define a compactification $M\cup M(\infty )$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M(\infty )$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M(\infty )$ for locally symmetric spaces. Moreover, $M(\infty )$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in the harmonic analysis of the locally symmetric space:1) it is the minimal Martin compactification for negative eigenvalues of the Laplacian, and 2) it can be used to parameterize the eigenfunctions of the Laplacian in continuous spectrum on $L_2$.},
affiliation = {University of Michigan, Department of Mathematics, Ann Arbor MI 48109-1003 (USA); Institute for Advanced Study, School of Mathematics, Princeton NJ 08540 (USA)},
author = {Ji, Lizhen, Macpherson, Robert},
journal = {Annales de l’institut Fourier},
keywords = {compactifications; locally symmetric spaces; geodesics; arithmetic groups},
language = {eng},
number = {2},
pages = {457-559},
publisher = {Association des Annales de l'Institut Fourier},
title = {Geometry of compactifications of locally symmetric spaces},
url = {http://eudml.org/doc/115986},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Ji, Lizhen
AU - Macpherson, Robert
TI - Geometry of compactifications of locally symmetric spaces
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 457
EP - 559
AB - For a locally symmetric space $M$, we define a compactification $M\cup M(\infty )$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M(\infty )$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M(\infty )$ for locally symmetric spaces. Moreover, $M(\infty )$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in the harmonic analysis of the locally symmetric space:1) it is the minimal Martin compactification for negative eigenvalues of the Laplacian, and 2) it can be used to parameterize the eigenfunctions of the Laplacian in continuous spectrum on $L_2$.
LA - eng
KW - compactifications; locally symmetric spaces; geodesics; arithmetic groups
UR - http://eudml.org/doc/115986
ER -

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