Spaces with nonpositive curvature and their ideal boundaries
A Hadamard space \(X\) is a simply connected metric space with nonpositive curvature in the sense of Alexandrov. If \(X\) is locally compact, then its ideal boundary \(\partial X\) is a compact metrizable space on which \(I(X)\), the isometry group of \(X\), acts by homeomorphisms. If \(X_1\) and \(X_2\) are in addition Gromov hyperbolic and \(f: X_1\to X_2\) is a quasi-isometry, then a) \(f\) induces a boundary homeomorphism \(\partial f:\partial X_1\to\partial X_2\). Moreover, b) if \(f\) is induced by a finitely generated group \(G\) that acts discretely, cocompactly and isometrically on both \(X_1\) and \(X_2\), then \(\partial f:\partial X_1\to\partial X_2\) is \(G\)-equivariant. The authors and Buyalo have shown that b) does not hold for arbitrary Hadamard spaces once the assumption of Gromov hyperbolicity is dropped. In this article, the authors show that a) does not hold for Hadamard spaces that are not Gromov hyperbolic. The main result is the following Theorem. There exists a pair of homeomorphic finite 2-complexes \(\overline{X_1}\) and \(\overline{X_2}\) with nonpositive curvature such that the ideal boundaries of the universal covers \(X_1\) and \(X_2\) are not homeomorphic. Let \(T_0\), \(T_1\) and \(T_2\) be flat 2-dimensional tori, and for \(i= 1,2\) let \(a_i\subseteq T_0\) and \(b_i\subseteq T_0\) be primitive closed geodesics with length \((a_i)= \text{length}(b_i)\). Assume furthermore that \(a_i\) and \(b_i\) lie in different free homotopy classes and meet exactly once at an angle \(\alpha\in (0,\pi/2]\). For \(i= 1,2\) glue \(T_i\) to \(T_0\) by identifying \(a_i\) and \(b_i\) isometrically. The resulting space \(\overline X(\alpha)\) is a length space with nonpositive curvature. If \(X(\alpha)\) denotes the universal cover of \(\overline X(\alpha)\), then the authors show that for \(0<\alpha< \pi/2\) the space \(\overline X(\alpha)\) is homeomorphic to \(\overline X(\pi/2)\) but \(X(\alpha)\) and \(X(\pi/2)\) have nonhomeomorphic ideal boundaries.
scientific article; zbMATH DE number 1436051