Summary:
We shall deal with two problems of Geometrical Group Theory which will be
developed into individual chapters.
A group G is called unsplittable if Hom(G,Z)=0 and this group is not a
non-trivial amalgam. Let X be a tree with a countable number of edges incident
at each vertex and G be its
automorphism group. In this paper we prove that the vertex
stabilizers are unsplittable groups.
Bass and Lubotzky proved (see [3]) that for certain
locally finite trees X, the automorphism group determines the tree X (that is,
knowing the automorphism group we can
``construct'' the tree X). We generalize this Theorem of Bass and Lubotzky,
using the above result. In particular we show that the Theorem holds even for
trees which are not locally finite.
Moreover, we prove that the permutation group of an infinite countable set is
unsplittable and the infinite (or finite) cartesian product of unsplittable
groups is an unsplittable group as well. Our results on this problem have led
to a publication (\cite{27}).
It should also be noted that recently Maciej Malicki provided some of our
results with a new proof} \cite{29}.
It is known that if the isoperimetric profile of a finite genus
non-compact surface grows faster than $\sqrt{t}$,
then it grows at least as a linear function.
In other words there are `gaps' in the isoperimetric profile of surfaces with
finite genus.
In this paper we show that no gap exists for surfaces of infinite genus. This
result of ours has been publicised on \cite{28}.
Keywords:
Trees, Unsplittable group, Rigidity theorem, Expander graph, Isoperimetric profile
