We study some algebraic properties of a class of group presentations depending on a finite number of integer parameters. This class contains many well-known groups which are interesting from a topological point of view. We find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3-manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family.

We study various questions about the generalised Fibonacci groups, a family of cyclically presented groups, which includes as special cases the Fibonacci, Sieradski, and Gilbert-Howie groups.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

Abstract. For a closed topological n-manifold X, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of X. In the case of a compact topological n-manifold with boundary (X, ∂X), the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In his classical book, Wall mentioned the possibility of introducing a mixed type of structure on a manifold with boundary. Following this suggestion, we introduce mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. The relationships can be described by using certain braids of exact sequences. Finally, we discuss explicitly several geometric examples.2000 Mathematics Subject Classification. Primary 57R67, 57Q10 Secondary 57R10, 55U35, 18F25.

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