286 research outputs found
L^2-Betti numbers of one-relator groups
We determine the L^2-Betti numbers of all one-relator groups and all
surface-plus-one-relation groups (surface-plus-one-relation groups were
introduced by Hempel who called them one-relator surface groups). In particular
we show that for all such groups G, the L^2-Betti numbers b_n^{(2)}(G) are 0
for all n>1. We also obtain some information about the L^2-cohomology of
left-orderable groups, and deduce the non-L^2 result that, in any
left-orderable group of homological dimension one, all two-generator subgroups
are free.Comment: 18 pages, version 3, minor changes. To appear in Math. An
On the concept of (homo)morphism : a key notion in the learning of abstract algebra
This article is dedicated to the investigation of difficulties involved in
the understanding of the homomorphism concept. It doesn't restrict to
group-theory but on the contrary raises the issue of developing teaching
strategies aiming at gaining access to structuralist thinking. Emphasis is put
on epistemological analysis and its interaction with didactics in an attempt to
make Abstract Algebra more accessible
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