59 research outputs found
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
Rips construction without unique product
Given a finitely presented group we produce a short exact sequence such that is a torsion-free
Gromov hyperbolic group without the unique product property and is without
the unique product property and has Kazhdan's Property (T). Varying we
show a wide diversity of concrete examples of Gromov hyperbolic groups without
the unique product property. As an application, we obtain Tarski monster groups
without the unique product property.Comment: 22 page
Graphical small cancellation groups with the Haagerup property
We prove the Haagerup property (= Gromov's a-T-menability) for finitely
generated groups defined by infinite presentations satisfying the graphical
C'(lambda)-small cancellation condition with respect to graphs endowed with a
compatible wall structure. We deduce that these groups are coarsely embeddable
into a Hilbert space and that the strong Baum-Connes conjecture and, hence, the
Baum-Connes conjecture with arbitrary coefficients hold for them. As the main
step we show that C'(lambda)-complexes satisfy the linear separation property.
Our result provides many new examples and a general technique to show the
Haagerup property for graphical small cancellation groups.Comment: 29 pages, minor modifications to v
Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms
We study the geometry of infinitely presented groups satisfying the small
cancelation condition C'(1/8), and define a standard decomposition (called the
criss-cross decomposition) for the elements of such groups. We use it to prove
the Rapid Decay property for groups with the stronger small cancelation
property C'(1/10). As a consequence, the Metric Approximation Property holds
for the reduced C*-algebra and for the Fourier algebra of such groups. Our
method further implies that the kernel of the comparison map between the
bounded and the usual group cohomology in degree 2 has a basis of power
continuum. The present work can be viewed as a first non-trivial step towards a
systematic investigation of direct limits of hyperbolic groups.Comment: 40 pages, 8 figure
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