an:01571175
Zbl 0989.20031
Kapovich, Michael; Kleiner, Bruce
Hyperbolic groups with low-dimensional boundary
EN
Ann. Sci. ??c. Norm. Sup??r. (4) 33, No. 5, 647-669 (2000).
00070378
2000
j
20F67 57M07 57M50
group boundaries; torsion-free hyperbolic groups; Menger curves; Sierpi??ski carpets; quasi-convex subgroups; hyperbolic Poincar?? duality groups
Authors' abstract: If a torsion-free hyperbolic group \(G\) has 1-dimensional boundary \(\partial_\infty G\), then \(\partial_\infty G\) is a Menger curve or a Sierpi??ski carpet provided \(G\) does not split over a cyclic group. When \(\partial_\infty G\) is a Sierpi??ski carpet we show that \(G\) is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincar?? duality group. We also construct a ``topologically rigid'' hyperbolic group \(G\): any homeomorphism of \(\partial_\infty G\) is induced by an element of \(G\).
J.W.Cannon (Provo)