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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The space of positive definite matrices and Gromov's invariant

Author: Richard P. Savage
Journal: Trans. Amer. Math. Soc. 274 (1982), 239-263
MSC: Primary 53C35; Secondary 53C20, 57R99
MathSciNet review: 670930
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Abstract: The space $ X_d^n{\text{of}}n \times n$ positive definite matrices with $ {\text{determinant}} = 1$ is considered as a subset of $ {{\mathbf{R}}^{n(n + 1)/2}}$ with isometries given by $ X \to AX{A^t}$ where the determinant of $ A = 1$ and $ X_d^n$ is given its invariant Riemannian metric. This space has a collection of simplices which are preserved by the isometries and formed by projecting geometric simplices in $ {{\mathbf{R}}^{n(n + 1)/2}}$ to the hypersurface $ X_d^n$. The main result of this paper is that for each $ n$ the volume of all top dimensional simplices of this type has a uniform upper bound.

This result has applications to Gromov's Invariant as defined in William P. Thurston's notes, The geometry and topology of $ 3$-manifolds. The result implies that the Gromov Invariant of the fundamental class of compact manifolds which are formed as quotients of $ X_d^n$ by discrete subgroups of the isometries is nonzero. This gives the first nontrivial examples of manifolds that have a nontrivial Gromov Invariant but do not have strictly negative curvature or nonvanishing characteristic numbers.

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Article copyright: © Copyright 1982 American Mathematical Society