The space of positive definite matrices and Gromov’s invariant

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.