Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

AbstractWe study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SLn(ℤ), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SLn(ℤ) for n≥ 3 cannot act on a visibility space X without fixing a point in $$\overline X $$ . Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.

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