Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings.

*(English)*Zbl 0910.53035An \((L,C)\) quasi-isometry is a map \(\Phi:X\to X^*\) between metric spaces such that for all points \(x,y\) in \(X\) we have \(L^{-1} d(x,y) -C\leq d(\Phi (x),\Phi (y)) \leq Ld(x,y)+C\) and \(d(x^*, \Phi (X)) <C\) for all \(x^*\) in \(X^*\). The idea of studying quasi-isometric spaces goes back to Marston Morse in the 20’s and has been used by many researchers since then, including Mostow in his proof of the strong rigidity theorem. Gromov was the first to study quasi-isometries in a systematic way, especially geometric properties of metric spaces that are preserved by quasi-isometries and the spaces that are quasi-isometric to a space of particular interest, for example, a Riemannian symmetric space with nonpositive sectional curvature.

It is relatively easy to find quasi-isometries between Euclidean, hyperbolic and complex hyperbolic spaces that do not preserve much of the geometry of these spaces. However, it is much more difficult to find quasi-isometries between the remaining symmetric spaces with strictly negative sectional curvature (rank 1). In the late 80’s, Pansu proved the following result: Theorem. Let \(X\) be either the quaternionic hyperbolic space \(\mathbb{H} H^n\) for \(n>1\) or the Cayley hyperbolic plane \(Ca H^2\). Then any quasi-isometry of \(X\) lies witin a bounded distance of an isometry of \(X\). Mostow proved this result in his strong rigidity theorem for quasi-isometries of \(X\) that are lifts of a homotopy equivalence between compact quotient manifolds \(X/ \Gamma \) and \(X/ \Gamma^*\), where \(\Gamma\) and \(\Gamma^*\) are discrete, cocompact groups of isometries of \(X\).

In the present work, the authors obtain some major results that are analogous of Pansu’s result for symmetric spaces of rank at least 2 with no Euclidean factor.

Theorem 1. For \(1\leq i\leq k\), \(1\leq j\leq k'\), let each \(X_i\), \(X_j'\) be either a nonflat irreducible symmetric space with nonpositive sectional curvature and no Euclidean factor or an irreducible thick Euclidean Tits building with cocompact affine Weyl group. Let \(X=\mathbb{R}^n\times X_1\times\cdots\times X_k\) and \(X^*= \mathbb{R}^{n^*} \times X^*_1 \times \cdots \times X^*_{k^*}\) be metric products. Then for every pair of positive numbers \(L,C\), there exist positive numbers \(L^*,C^*\) and \(D^*\) such that the following holds. Let \(\Phi: X\to X^*\) be an \((L,C)\) quasi-isometry. Then \(n=n^*\), \(k=k^*\) and, after reindexing the factors of \(X^*\), there are \((L^*,C^*)\) quasi-isometries \(\Phi_i: X_i\to X^*_i\) so that \(d(p^*\circ \Phi,(\Phi_1 \times\cdots \times\Phi_k) \circ p) <D\), where \(p:X \to X_1 \times \cdots \times X_k\) and \(p^*: X^* \to X^*_1 \times \cdots \times X^*_k\) are the projections.

Theorem 2. Let \(X\) and \(X^*\) be as in Theorem 1, but assume, in addition, that \(X\) is either a nonflat irreducible symmetric space of nonpositive sectional curvature and rank at least 2 or a thick irreducible Euclidean building of rank at least 2 with cocompact affine Weyl group and Moufang Tits boundary. Then any \((L,C)\) quasi-isometry \(\Phi:X\to X^*\) lies at distance \(<D\) from a homothety \(\Phi_0: X\to X^*\), where \(D\) depends only on \(L\) and \(C\).

The authors and Kapovich have strengthened Theorem 1 in another article. Theorem 2 settles a conjecture of Mostow from the 70’s, and Leeb has shown subsequently that the Moufang condition in Theorem 2 can be dropped. As a corollary of Theorems 1 and 2, the authors obtain a classification of symmetric spaces with nonpositive sectional curvature and no Euclidean factor up to quasi-isometry: Corollary. Let \(X\) and \(X^*\) be symmetric spaces with nonpositive sectional curvature and no Euclidean factor. If \(X\) and \(X^*\) are quasi-isometric, then they become isometric after multiplying the metrics on de Rham factors by suitable positive constants.

This article presents a self-contained exposition of buildings, but reformulated for convenience in terms of metric geometry. An essential ingredient is also a discussion of nonpositively curved metric spaces that are not locally compact (and hence not manifolds).

It is relatively easy to find quasi-isometries between Euclidean, hyperbolic and complex hyperbolic spaces that do not preserve much of the geometry of these spaces. However, it is much more difficult to find quasi-isometries between the remaining symmetric spaces with strictly negative sectional curvature (rank 1). In the late 80’s, Pansu proved the following result: Theorem. Let \(X\) be either the quaternionic hyperbolic space \(\mathbb{H} H^n\) for \(n>1\) or the Cayley hyperbolic plane \(Ca H^2\). Then any quasi-isometry of \(X\) lies witin a bounded distance of an isometry of \(X\). Mostow proved this result in his strong rigidity theorem for quasi-isometries of \(X\) that are lifts of a homotopy equivalence between compact quotient manifolds \(X/ \Gamma \) and \(X/ \Gamma^*\), where \(\Gamma\) and \(\Gamma^*\) are discrete, cocompact groups of isometries of \(X\).

In the present work, the authors obtain some major results that are analogous of Pansu’s result for symmetric spaces of rank at least 2 with no Euclidean factor.

Theorem 1. For \(1\leq i\leq k\), \(1\leq j\leq k'\), let each \(X_i\), \(X_j'\) be either a nonflat irreducible symmetric space with nonpositive sectional curvature and no Euclidean factor or an irreducible thick Euclidean Tits building with cocompact affine Weyl group. Let \(X=\mathbb{R}^n\times X_1\times\cdots\times X_k\) and \(X^*= \mathbb{R}^{n^*} \times X^*_1 \times \cdots \times X^*_{k^*}\) be metric products. Then for every pair of positive numbers \(L,C\), there exist positive numbers \(L^*,C^*\) and \(D^*\) such that the following holds. Let \(\Phi: X\to X^*\) be an \((L,C)\) quasi-isometry. Then \(n=n^*\), \(k=k^*\) and, after reindexing the factors of \(X^*\), there are \((L^*,C^*)\) quasi-isometries \(\Phi_i: X_i\to X^*_i\) so that \(d(p^*\circ \Phi,(\Phi_1 \times\cdots \times\Phi_k) \circ p) <D\), where \(p:X \to X_1 \times \cdots \times X_k\) and \(p^*: X^* \to X^*_1 \times \cdots \times X^*_k\) are the projections.

Theorem 2. Let \(X\) and \(X^*\) be as in Theorem 1, but assume, in addition, that \(X\) is either a nonflat irreducible symmetric space of nonpositive sectional curvature and rank at least 2 or a thick irreducible Euclidean building of rank at least 2 with cocompact affine Weyl group and Moufang Tits boundary. Then any \((L,C)\) quasi-isometry \(\Phi:X\to X^*\) lies at distance \(<D\) from a homothety \(\Phi_0: X\to X^*\), where \(D\) depends only on \(L\) and \(C\).

The authors and Kapovich have strengthened Theorem 1 in another article. Theorem 2 settles a conjecture of Mostow from the 70’s, and Leeb has shown subsequently that the Moufang condition in Theorem 2 can be dropped. As a corollary of Theorems 1 and 2, the authors obtain a classification of symmetric spaces with nonpositive sectional curvature and no Euclidean factor up to quasi-isometry: Corollary. Let \(X\) and \(X^*\) be symmetric spaces with nonpositive sectional curvature and no Euclidean factor. If \(X\) and \(X^*\) are quasi-isometric, then they become isometric after multiplying the metrics on de Rham factors by suitable positive constants.

This article presents a self-contained exposition of buildings, but reformulated for convenience in terms of metric geometry. An essential ingredient is also a discussion of nonpositively curved metric spaces that are not locally compact (and hence not manifolds).

Reviewer: P.Eberlein (Chapel Hill)

##### MSC:

53C35 | Differential geometry of symmetric spaces |

##### Keywords:

quasi-isometric classification of symmetric spaces; Mostow rigidity theorem; Euclidean buildings; curvature##### References:

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