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A Note on Shortest Developments
De Vrijer has presented a proof of the finite developments theorem which, in
addition to showing that all developments are finite, gives an effective
reduction strategy computing longest developments as well as a simple formula
computing their length.
We show that by applying a rather simple and intuitive principle of duality
to de Vrijer's approach one arrives at a proof that some developments are
finite which in addition yields an effective reduction strategy computing
shortest developments as well as a simple formula computing their length. The
duality fails for general beta-reduction.
Our results simplify previous work by Khasidashvili
Linear sofic groups and algebras
We introduce and systematically study linear sofic groups and linear sofic
algebras. This generalizes amenable and LEF groups and algebras. We prove that
a group is linear sofic if and only if its group algebra is linear sofic. We
show that linear soficity for groups is a priori weaker than soficity but
stronger than weak soficity. We also provide an alternative proof of a result
of Elek and Szabo which states that sofic groups satisfy Kaplansky's direct
finiteness conjecture.Comment: 34 page
Infinite groups with fixed point properties
We construct finitely generated groups with strong fixed point properties.
Let be the class of Hausdorff spaces of finite covering
dimension which are mod- acyclic for at least one prime . We produce the
first examples of infinite finitely generated groups with the property that
for any action of on any , there is a global fixed
point. Moreover, may be chosen to be simple and to have Kazhdan's property
(T). We construct a finitely presented infinite group that admits no
non-trivial action by diffeomorphisms on any smooth manifold in
. In building , we exhibit new families of hyperbolic
groups: for each and each prime , we construct a non-elementary
hyperbolic group which has a generating set of size , any proper
subset of which generates a finite -group.Comment: Version 2: 29 pages. This is the final published version of the
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