Agnese Ilaria Telloni

Combinatorics of a class of groups with cyclic presentation

We study a family of combinatorial closed 3-manifolds obtained from polyhedral 3-balls, whose finitely many boundary faces are glued together in pairs. Then we determine geometric presentations of their fundamental groups, and find conditions under which such groups are infinite and/or aspherical. Moreover, we show that our presentations are a natural generalization of those considered by Prishchepov in [M.I. Prishchepov, Asphericity, atoricity and symmetrically presented groups, Comm. Algebra 23 (13) (1995) 5095-5117]. Finally we illustrate some geometric and topological properties of the constructed manifolds, as, for example, a combinatorial description of them as cyclic coverings of the 3-sphere branched over some specified classes of knots.

Cyclic Branched Coverings Over Some Classes of (1,1)-Knots

We construct a 4-parametric family of combinatorial closed 3-manifolds, obtained by glueing together in pairs the boundary faces of polyhedral 3-balls. Then, we obtain geometric presentations of the fundamental groups of these manifolds and determine the corresponding split extension groups. Finally, we prove that the considered manifolds are cyclic coverings of the 3-sphere branched over well-specified -knots, including torus knots and Montesinos knots.

Dehn surgeries on some classical links

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

Some Hyperbolic 3-Manifolds with Totally Geodesic Boundary

We construct a family of compact hyperbolic 3-manifolds with totally geodesic boundary, depending on three integer parameters. Then we determine geometric presentations of the fundamental groups of these manifolds and prove that they are cyclic coverings of the 3-ball branched along a specified tangle with two components. Finally, we give a classification of these manifolds up to homeomorphism (resp., isometry), and determine their isometry groups.

Isometry Groups of Some Dunwoody Manifolds

Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly 𝕊3) branched over (1,1)-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.

SOME HYPERBOLIC SPACE FORMS WITH FEW GENERATED FUNDAMENTAL GROUPS

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links.

On the groups of some fibered spaces

Abstract We introduce a family of cyclically presented groups defined by five parameters, suggested by fundamental groups of certain closed 3-manifolds. This family includes groups studied by Sidki in [On the fundamental groups of 3-manifolds of Lins–Mandel]. We determine several algebraic properties of our family of groups, and give a geometrical interpretation of such groups for particular choices of parameters.