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• W2991440020 abstract "A group G is called subgroup conjugacy separable (SCS) if for any two f.g. non-conjugate subgroups H and H' of G, there exists a finite quotient E of G such that the images of H and H' remain non-conjugate in E. In this thesis we will prove that, that any finitely generated virtually free group is SCS. This proof uses essentially covering theory of graphs of groups. Note that the introduction and the elaboration of this covering theory will be done as well in this thesis. As a direct consequence of the established formula for counting the index of a finite index subgroup of the fundamental group of a graph of groups, we receive a formula for counting the rank of a free finite index subgroup of a virtually free group in terms of the index of the corresponding subgroup and the finite orders of the vertex groups of the original finite graph of finite groups. For doing this we make use of the well-known fact that any f.g. virtually free group splits as a finite graph of finite groups. Furthermore in the context of covering theory of graphs of groups, we will give a reasonable definition for the group of Decktransformations associated to a cover of a graph of groups and we will establish a connection between this group and the normalizer of the represented subgroup. Later on the covering theory of graphs of groups enables us to prove the subgroup separability for some special kinds of towers over a finitely generated virtually-free group. Note that a group G is called subgroup separable if for any subgroup U of G and any element g of G, which does not lie in U, there exists a finite index subgroup H of G containing U satisfying that g does not lie in H. Afterwards we are going to show that Fuchsian Groups are SCS. This proof uses essentially covering theory of 2-dimensional orbifolds. Last but not least we will use this statement to show that the fundamental group of any totally orientable Seifert fibered space is SCS.%%%%Eine Gruppe G ist untergruppenkonjugationsseparabel (SCS), falls fur je zwei endlich erzeugte, nicht konjugierte Untergruppen H und H' von G ein endlicher Quotient E von G existiert, s.d. die Bilder von H und H' in E nicht konjugiert sind. Wir werden in dieser Arbeit einen Beweis dafur liefern, dass endlich erzeugte virtuell freie Gruppen SCS sind. Dieser benutzt in erster Linie Uberlagerungstheorie von Graphen von Gruppen, welche in dieser Arbeit auch ausfuhrlich eingefuhrt und ausgeabreitet wird. Als eine direkte Konsequenz der so gewonnenen Formel fur den Index einer von einer Uberlagerung von Graphen von Gruppen reprasentierten Untergruppe erhielten wir eine Formel fur den Rang einer freien Unterguppe von endlichem Index in einer endlich erzeugten virtuell freien Gruppe. Dafur nutzten wir die Tatsache aus, dass jede endlich erzeugte virtuell freie Gruppe als Fundamentalgruppe eines endlichen Graphen von endlichen Gruppen aufgefasst werden kann. Des Weiteren werden wir im Sinne der erarbeiteten Uberlagerungstheorie von Graphen von Gruppen eine sinnvolle Definition der Decktransformationsgruppe einer…" @default.
• W2991440020 created "2019-12-05" @default.
• W2991440020 creator A5028040415 @default.
• W2991440020 date "2018-04-16" @default.
• W2991440020 modified "2023-09-26" @default.
• W2991440020 title "Separability Properties and Finite-Sheeted Coverings of Graphs of Groups and 2-dimensional Orbifolds" @default.
• W2991440020 hasPublicationYear "2018" @default.
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