The space of positive definite matrices and Gromov’s invariant
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- by Richard P. Savage PDF
- Trans. Amer. Math. Soc. 274 (1982), 239-263 Request permission
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.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 274 (1982), 239-263
- MSC: Primary 53C35; Secondary 53C20, 57R99
- DOI: https://doi.org/10.1090/S0002-9947-1982-0670930-8
- MathSciNet review: 670930