Finite permutation groups of rank 3

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46298/1/209_2005_Article_BF01111335.pd

On finite affine planes of rank 3

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46256/1/209_2005_Article_BF01109877.pd

Coherent algebras

Coherent algebras are defined to be the subalgebras of the matrix algebras Mn() closed under Hadamard ( = coefficientwise) multiplication and containing the all 1 matrix, and are shown to be precisely the adjacency algebras of coherent configurations. Each such algebra has a type, which is a symmetric matrix with positive integer entries. The theory is illustrated by applications to quasisymmetric designs, which are essentially equivalent to coherent algebras of type .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26620/1/0000161.pd

Characterization of families of rank 3 permutation groups by the subdegrees I

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46626/1/13_2005_Article_BF01220896.pd

On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio

Characterization of families of rank 3 permutation groups by the subdegrees II

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46712/1/13_2005_Article_BF01220928.pd

Largeness and SQ-universality of cyclically presented groups

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems