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T T T T T T T T T T T T T T T Abstract We provide, for hyperbolic and flat 3–manifolds, obstructions to bounding hy-perbolic 4–manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.

Let d be a square free positive integer and O d the ring of integers in Q(√ −d). The main result of this paper is that the groups PSL(2, O d) are subgroup separable on geometrically finite subgroups.

We discuss separability properties of discrete groups, and introduce a new property of groups that is motivated by a geometric proof of separability of geometrically finite subgroups of Kleinian groups. This property appears natural in that it provides a general framework for old questions in the geometry and topology of hyperbolic manifolds and discrete… (More)

If Γ is a finite co-area Fuchsian group acting on H 2 , then the quotient H 2 /Γ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a finite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and… (More)

- C Mca, Gordon, D D Long, A W Reid, E M Friedlander
- 2003

We prove that any Coxeter group that is not virtually free contains a surface group. In particular if the Coxeter group is word hyperbolic and not virtually free this establishes the existence of a hyperbolic surface group, and answers in the aarmative a question of Gromov in this setting. We also discuss when Artin groups contain hyperbolic surface groups.

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property τ. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.