Mid Sweden University

We prove that any Lie subgroup G (with finitely many connected components) of an infinite-dimensional topological group (Formula presented.) which is an amalgamated product of two closed subgroups can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).

We study the thickening problem on a 2-dimensional Stein variety X with isolated irreducible singularities, i.e. the problem whether the assumption that a compact set K is contained in the interior of another compact set L implies that the same inclusion holds for their holomorphic hulls. The problem is still open except for a positive answer in the special case of quotient singularities. The main result of the present article is the partial result that the holomorphic hull L is has a directional density property at every singular point contained in the hull of K. The proof is based on removability results on pseudoconcave closed sets, which may be of some independent interest.

On a normal Stein variety X, we study the thickening problem, i.e. the problem whether the assumption that a compact set K is contained in the interior of another compact set, L implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for X with isolated quotient singularities. On any Stein space X with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension 2, we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries.

We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group, which is known to be dense in the identity component of the holomorphic automorphism group, is a free product.

We define the notion of shears and overshears on a Danielewski surface. We show that the group generated by shears and overshears is dense (in the compact open topology) in the path-connected component of the identity of the holomorphic automorphism group.

In this thesis we define the notion of an overshear on a Danielewskisurface. Next we show that the group generated by the overshears is dense in the component of the identity of the automorphism group. Moreover, we show that the overshear group has a structure of an amalgamated product, and as consequence of this the overshear group is a proper subgroup of the automorphism group. Finally we classify the R^n-actions, and therefore the one parameter subgroups, of the overshear group. We also show that any Lie subgroup of an amalgamated product can be conjugated to one of the factors of the amalgamated product.

In this thesis we define the notion of an overshear on a Danielewski surface. We show that the group generated by the overshears is dense (in the compact open topology) in the automorphism group for small degrees of the defining polynomial. It is also shown in the thesis that the overshear group has a structure of an amalgamated product. Finally, we show that the Danielewski surfaces have the Oka-Grauert property.