Expanders and k-theory for group c* algebras

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Paul Baum , Penn State University
Fine Hall 214

An expander or expander family is a sequence of finite graphs X_1, X_2, X_3,... which is efficiently connected.   A discrete group $G$ which contains an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. Such a group is known as the Gromov monster and is the only known example of a non-exact group.  The left side of BC with coefficients  sees" any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also sees" any group as if the group were exact. For exact groups (i.e. all groups except the Gromov  monster) there is no change in BC with coefficients. In the corrected form of BC with coefficients the Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (April, 2014) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Guentner and R. Willett.