On twisted l^2-Betti numbers
20.30 SR 2.058
l^2-Betti numbers were originally defined as the analogous of normal Betti numbers for G-CW-complexes for a group G. They measure the von Neumann dimension of the cellular homology groups of a finite type free G-CW-complex with local coefficients in l^2(G). Wolfgang Lück introduces twisted l^2-Betti numbers by twisting with a C[G]-module M that is finite-dimensional as a C-vector space. A more algebraic way to study l^2-Betti numbers is to study Sylvester matrix rank functions on C[G]. In this talk we will explain how to twist a matrix A in Mat_n(C[G]) with a finite dimensional representation of G with values in GL_m(C). This gives us a new matrix B in Mat_mn(C[G]). We will show that when G is a sofic group rk_G(B) =m rk_G(A). This is a joint work with Andrei Jaikin.