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Involutions of negatively curved groups with wild boundary behavior. (English) Zbl 1189.57014

An embedding \(S^n \hookrightarrow S^{n+2}\) is said to be tame at a point \(p\in S^n\) if there exists a neighborhood \(U\) of \(p\) in \(S^{n+2}\) with the property that \((U,S^n \cap U)\) is homeomorphic to \((\mathbb R^{ n+2}, \mathbb R^n)\). A point \(p\in S^n\) that is not tame is said to be wild. An embedding is tame provided it is tame at every point, and it is totally wild if it is wild at every point. Examples of non-tame embeddings of 2-spheres in 3-spheres include the Alexander horned sphere, the Antoine wild sphere and the Fox-Artin ball. In the paper under review the authors prove that if \(\Gamma\) is a \(\delta\)-hyperbolic group and \(\Lambda \leq \Gamma \) a quasi-convex subgroup and we suppose that \( \partial ^{\infty}\Gamma = S^{n+2}_{\infty} , \partial ^{\infty} \Lambda = S^n_{\infty} \) where \(n\neq 0,3\) and that the embedding \(S^n_{\infty} \hookrightarrow S^{n+2}_{\infty}\) induced by the inclusion \(\Lambda \leq \Gamma\) is knotted, then the embedding is a totally wild knot. The authors also give some examples of such embeddings. Also, as a corollary of the previous result they prove a Generalized Smith Conjecture for algebraic involutions, namely, if \(\tau: S^{n+2} \rightarrow S^{n+2}\) is an algebraic self-homeomorphism (\(n\not= 0,3\)) of finite order, and the fixed point set is a tame embedded \(S^n\hookrightarrow S^{n+2}\), then such an embedding is unknotted, where by an algebraic self-homeomorphism of finite order they mean an algebraic self-homeomorphism induced by an automorphism of finite order.

MSC:

57M30 Wild embeddings
20F67 Hyperbolic groups and nonpositively curved groups
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