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 positive definite matrices with is considered as a subset of with isometries given by where the determinant of and 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 to the hypersurface . The main result of this paper is that for each 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* *-manifolds*. The result implies that the Gromov Invariant of the fundamental class of compact manifolds which are formed as quotients of 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.

**[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**-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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DOI:
https://doi.org/10.1090/S0002-9947-1982-0670930-8

Article copyright:
© Copyright 1982
American Mathematical Society