P. Baum, N. Higson, and R. Plymen :
“Equivariant homology for \( \mathrm{SL}(2) \) of a \( p \) -adic field ,”
pp. 1–18
in
Index theory and operator algebras
(Boulder, CO, 6–10 August 1991 ).
Edited by J. Fox and P. Haskell .
Contemporary Mathematics 148 .
American Mathematical Society (Providence, RI ),
1993 .
MR
1228497
Zbl
0844.46043
incollection

Abstract
People
BibTeX

Let \( F \) be a \( p \) -adic field and let \( G = SL(2) \) be the group of unimodular \( 2{\times}2 \) matrices over \( F \) . The aim of this paper is to calculate certain equivariant homology groups attached to the action of \( G \) on its tree. They arise in connection with a theorem of M. Pimsner on the \( K \) -theory of the \( C^* \) -algebra of \( G \) [1986], and our purpose is to explore the representation theoretic content of Pimsner’s result.

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AUTHOR = {Baum, Paul and Higson, Nigel and Plymen,
Roger},
TITLE = {Equivariant homology for \$\mathrm{SL}(2)\$
of a \$p\$-adic field},
BOOKTITLE = {Index theory and operator algebras},
EDITOR = {Fox, J. and Haskell, P.},
SERIES = {Contemporary Mathematics},
NUMBER = {148},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1993},
PAGES = {1--18},
DOI = {10.1090/conm/148/01246},
NOTE = {(Boulder, CO, 6--10 August 1991). MR:1228497.
Zbl:0844.46043.},
ISSN = {0271-4132},
ISBN = {9780821851524},
}
P. Baum, A. Connes, and N. Higson :
“Classifying space for proper actions and \( K \) -theory of group \( C^* \) -algebras ,”
pp. 241–291
in
\( C^* \) -algebras: 1943–1993
(San Antonio, TX, 13–14 January 1993 ).
Edited by R. S. Doran .
Contemporary Mathematics 167 .
American Mathematical Society (Providence, RI ),
1994 .
MR
1292018
Zbl
0830.46061
incollection

Abstract
People
BibTeX

We announce a reformulation of the conjecture in [1982; 1988]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [1988]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.

@incollection {key1292018m,
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Nigel},
TITLE = {Classifying space for proper actions
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BOOKTITLE = {\$C^*\$-algebras: 1943--1993},
EDITOR = {Doran, Robert S.},
SERIES = {Contemporary Mathematics},
NUMBER = {167},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1994},
PAGES = {241--291},
DOI = {10.1090/conm/167/1292018},
NOTE = {(San Antonio, TX, 13--14 January 1993).
MR:1292018. Zbl:0830.46061.},
ISSN = {0271-4132},
ISBN = {9780821851753},
}
P. Baum, N. Higson, and R. Plymen :
Cosheaf homology and \( K \) theory for \( p \) -adic groups .
Preprint ,
Pennsylvania State University ,
1995 .
techreport

People
BibTeX
@techreport {key25400440,
AUTHOR = {Baum, Paul and Higson, N. and Plymen,
R.},
TITLE = {Cosheaf homology and \$K\$ theory for
\$p\$-adic groups},
TYPE = {preprint},
INSTITUTION = {Pennsylvania State University},
YEAR = {1995},
}
P. Baum, N. Higson, and R. Plymen :
“A proof of the Baum–Connes conjecture for \( p \) -adic \( \mathrm{GL}(n) \) ,”
C. R. Acad. Sci. Paris Sér. I Math.
325 : 2
(July 1997 ),
pp. 171–176 .
MR
1467072
Zbl
0918.46061
article

Abstract
People
BibTeX
@article {key1467072m,
AUTHOR = {Baum, Paul and Higson, Nigel and Plymen,
Roger},
TITLE = {A proof of the {B}aum--{C}onnes conjecture
for \$p\$-adic \$\mathrm{GL}(n)\$},
JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {325},
NUMBER = {2},
MONTH = {July},
YEAR = {1997},
PAGES = {171--176},
DOI = {10.1016/S0764-4442(97)84594-6},
NOTE = {MR:1467072. Zbl:0918.46061.},
ISSN = {0764-4442},
}
P. F. Baum, N. Higson, and R. J. Plymen :
“Representation theory of \( p \) -adic groups: A view from operator algebras ,”
pp. 111–149
in
The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis
(Baltimore, MD, 9–10 January 1998 ).
Edited by R. Doran and V. Varadarajan .
Proceedings of Symposia in Pure Mathematics 68 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1767895
Zbl
0982.19006
incollection

Abstract
People
BibTeX
@incollection {key1767895m,
AUTHOR = {Baum, P. F. and Higson, N. and Plymen,
R. J.},
TITLE = {Representation theory of \$p\$-adic groups:
{A} view from operator algebras},
BOOKTITLE = {The mathematical legacy of {H}arish-{C}handra:
{A} celebration of representation theory
and harmonic analysis},
EDITOR = {Doran, R. and Varadarajan, V.},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {68},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {111--149},
DOI = {10.1090/pspum/068/1767895},
NOTE = {(Baltimore, MD, 9--10 January 1998).
MR:1767895. Zbl:0982.19006.},
ISSN = {0082-0717},
ISBN = {9780821811979},
}
P. Baum, N. Higson, and T. Schick :
“On the equivalence of geometric and analytic \( K \) -homology ,”
pp. 1–24
in
Special issue: In honor of Robert D. MacPherson, Part 3 ,
published as Pure Appl. Math. Q.
3 : 1 .
International Press (Sommerville, MA ),
2007 .
MR
2330153
Zbl
1146.19004
incollection

Abstract
People
BibTeX
@article {key2330153m,
AUTHOR = {Baum, Paul and Higson, Nigel and Schick,
Thomas},
TITLE = {On the equivalence of geometric and
analytic \$K\$-homology},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {3},
NUMBER = {1},
YEAR = {2007},
PAGES = {1--24},
DOI = {10.4310/PAMQ.2007.v3.n1.a1},
NOTE = {\textit{Special issue: {I}n honor of
{R}obert {D}. {M}ac{P}herson, Part 3}.
MR:2330153. Zbl:1146.19004.},
ISSN = {1558-8599},
}
P. Baum, N. Higson, and T. Schick :
“A geometric description of equivariant \( K \) -homology for proper actions ,”
pp. 1–22
in
Quanta of maths: Proceedings of meeting in honor of Alain Connes’ 60th birthday .
Edited by E. Blanchard, D. Ellwood, M. Khalkhali, M. Marcolli, H. Moscovici, and S. Popa .
Clay Mathematics Proceedings 11 .
American Mathematical Society (Providence, RI ),
2010 .
MR
2732043
Zbl
1216.19006
ArXiv
0907.2066
incollection

Abstract
People
BibTeX

Let \( G \) be a discretre group and let \( X \) be a \( G \) -finite, proper \( G \) -CW-complex. We prove that Kasparov’s equivariant \( K \) -homology groups
\[ KK_*^G(C_0(X),\mathbb{C}) \]
are isomorphic to the geometric equivariant \( K \) -homology groups of \( X \) that are obtained by making the geometric \( K \) -homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum–Connes conjecture for discrete groups.

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AUTHOR = {Baum, Paul and Higson, Nigel and Schick,
Thomas},
TITLE = {A geometric description of equivariant
\$K\$-homology for proper actions},
BOOKTITLE = {Quanta of maths: {P}roceedings of meeting
in honor of {A}lain {C}onnes' 60th birthday},
EDITOR = {Blanchard, Etienne and Ellwood, David
and Khalkhali, Masoud and Marcolli,
Matilde and Moscovici, Henri and Popa,
Sorin},
SERIES = {Clay Mathematics Proceedings},
NUMBER = {11},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2010},
PAGES = {1--22},
NOTE = {ArXiv:0907.2066. MR:2732043. Zbl:1216.19006.},
ISSN = {1534-6455},
ISBN = {9780821852033},
}