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Hyperbolic manifolds, amalgamated products and critical exponents

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Title: Hyperbolic manifolds, amalgamated products and critical exponents
Alternate Title: Varie´te´s hyperboliques, produits amalgame´s et exposants critiques
Authors: Besson, Gérard1 G.Besson@fourier.ujf-grenoble.fr, Courtois, Gilles2 courtois@math.polytechnique.fr, Gallot, Sylvestre1 Sylvestre.Gallot@fourier.ujf-grenoble.fr
Source: Comptes Rendus. Mathématique. Feb2003, Vol. 336 Issue 3, p257. 5p.
Subjects: Hyperbolic spaces, Curvature
Abstract (English): We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of dimn is a free product of its subgroups A and B over the amalgamated subgroup C, then the critical exponent of C is not smaller than n−2. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]
Abstract (French): Nous donnons une preuve nouvelle d'un re´sultat duˆ a` Y. Shalom ; pre´cise´ment, nous montrons que, si le groupe fondamental d'une varie´te´ hyperbolique re´elle compacte de dimension n est le produit libre de ses sous-groupes A et B amalgame´ sur C, alors l'exposant critique de C est plus grand que n−2. La preuve, ge´ome´trique, permet de traiter le cas d'e´galite´ ainsi qu'une extension au cas de courbure variable. Pour citer cet article : G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright 2003 Elsevier]
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Database: Engineering Source
Description
Abstract:We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of <f>dimn</f> is a free product of its subgroups <f>A</f> and <f>B</f> over the amalgamated subgroup <f>C</f>, then the critical exponent of <f>C</f> is not smaller than <f>n−2</f>. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]
ISSN:1631073X
DOI:10.1016/S1631-073X(02)00019-5