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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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  • [1] Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 0146301
  • [2] C. Carathéodory, Ueber den Variabilitatsbereich der Fourierschen Konstanten von Positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217.
  • [3] Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
  • [4] Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
  • [5] Morris W. Hirsch and William P. Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. Math. (2) 101 (1975), 369–390. MR 0370615
  • [6] John Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215–223. MR 0095518
  • [7] Dennis Sullivan, A generalization of Milnor’s inequality concerning affine foliations and affine manifolds, Comment. Math. Helv. 51 (1976), no. 2, 183–189. MR 0418119
  • [8] W. Thurston, The geometry and topology of $ 3$-manifolds, Princeton Univ. Press, Princeton, N. J., 1978.
  • [9] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 0286898

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Article copyright: © Copyright 1982 American Mathematical Society