Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Oleg Bogopolski
Universität Düsseldorf
Generalized presentations of infinite groups, in particular of Aut(Fω).
A very common way to describe a group G is by specifying generators and relations for G, with other
words, by finding a set Λ and a subset R of the free group F(Λ) such that G is isomorphic to the quotient
of F(Λ) by the normal closure 〈〈R〉〉 of R. Unfortunately, for certain groups G, it is difficult to find such a
presentation.
For example, take an infinite set X and look at the symmetric group Σ(X) of all permutations of X. It
is obvious that the set of transpositions does not generate Σ(X). However, it is easy to see that each
element of Σ(X) is, in a certain sense, an infinite product of transpositions. In the present paper, we are
going to develop the concept of a generalized presentation of a group G. In the case of Σ(X), generalized
generators are just the transpositions, and generalized relations are exactly the relations familiar from the
finite symmetric groups Σn.
An important ingredient of a generalized presentation is the notion of a big free group BF(Λ). If Λ is
countably infinite, the group BF(Λ) was first studied by Higman in 1952; it is isomorphic to the fundamental
group of the Hawaiian earrings. Whereas subgroups of free groups are again free, subgroups of big free
groups need not be big free groups. Therefore we have to introduce the concept of generalized free
groups; these are certain subgroups of big free groups.
A generalized presentation of a group G consists, then, of a generalized free group F and of a subset R
of F such that G is isomorphic, not simply to F /〈〈R〉〉, but to the quotient of F by the closure of 〈〈R〉〉 with
respect to an appropriate topology on the group F .
We give generalized presentations of the symmetric group Σ(X) and of the automorphism group of the
free group of infinite countable rank, Aut(Fω). This is a joint work with Wilhelm Singhof.
Kai-Uwe Bux
Universität Bielefeld
Subgroup conjugacy separability of surface groups
Let G be a group. We call two elements g and h conjugacy separated if there is a finite quotient Q of
G where g and h have non-conjugate images. We say that G is conjugacy separable if any two nonconjugate
elements are conjugacy separated. Taking h = 1, we see that conjugacy separability implies
residual finiteness.
Similarly, we call subgroups H and H ′ of G conjugacy separated if there is a finite quotient Q of G where
H and H ′ have non-conjugate images; and we say that G is subgroup conjugacy separable (SCS) if any
two non-conjugate finitely generated subgroups are conjugacy separated. This property can be viewed
as the subgroup-analogue of conjugacy separability very much in the same spirit as LERF is a subgroupanalogue
of residual finiteness.
Bogopolski and Grunewald have shown that free groups of finite rank are SCS. We show that fundamental
groups of closed orientable surfaces are SCS. We conjecture that limit groups are SCS (they have been
shown to be LERF).
In the talk, I shall outline the proof for fundamental groups of genus at least 2. In particular, I want to
stress how the hyperbolic metric that such a surface admits enters the picture. This is a joint work with
Oleg Bogopolski.
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