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Infinite index subgroups and finiteness properties of intersections of geometrically finite groups. (English) Zbl 1139.57033

Summary: We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup. We produce several examples of such intersections of geometrically finite groups including finitely generated but not finitely presented discrete subgroups.

MSC:

57S25 Groups acting on specific manifolds
20F65 Geometric group theory
53C35 Differential geometry of symmetric spaces
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