arXiv subjects
Publications and source records attributed to Bakul Sathaye.
We show that every group acting freely and vertex-transitively by isometries on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the visual boundary homeomorphic to a join of two copies of the Cantor set.
A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpiński's theorem on embedding $1$--dimensional planar compacta into the Sierpiński carpet. See the appendix for a brief erratum.
We extend the methods of Davis-Januszkiewicz-Lafont to provide a new obstruction to smooth Riemannian metric with non-positive sectional curvature. We construct examples of locally CAT(0) 4-manifolds $M$, whose universal covers satisfy isolated flats condition and contain 2-dimensional flats with the property that $\sqcup\ \partial^\infty F_i \hookrightarrow \partial^\infty \tilde M$ are non-trivial links that are not isotopic to any great circle link. Further, all the flats in $\tilde M$ are unknotted at infinity, and yet $M$ does not have a Riemannian smoothing.
Given an $m$-component link $L$ in $S^3$ ($m \ge 2$), we construct a family of links which are link homotopic, but not link isotopic, to $L$. Every proper sublink of such a link is link isotopic to the corresponding sublink of $L$. Moreover, if $L$ is an unlink then there exist links that in addition to the above properties have all Milnor invariants zero.