Ph.D Thesis | |

Ph.D Student | Meir Ben Efraim Ehud |
---|---|

Subject | On Certain Cohomological Properties of a Group and their Reflection in a Finite Index Subgroup |

Department | Department of Mathematics |

Supervisors | PROF. Eli Aljadeff |

PROF. Shlomo Gelaki |

This thesis deals with relations between cohomological properties of a group and a finite index subgroup.

The first chapter deals with a
conjecture of Moore. Moore's conjecture states that if *G* is a group and *H*
a finite index subgroup of *G*, then under a certain natural assumption on
the embedding of *H* in *G*, every *G* module which is
projective over *H* is projective over *G*. Moore's conjecture generalizes
a theorem of Serre about cohomological dimension of torsion free groups. It
also generalizes Chouinard's theorem for finite groups, which states that if *G*
is a finite group, then a *G* module *M* which is projective over
every elementary abelian subgroup of *G* is projective over *G*. It
was known that the conjecture is true for finite groups and groups of finite
cohomlogical dimension. In this thesis we generalize these results and prove
that the conjecture is true for all the groups in Kropholler's hierarchy *LHF*.
We also show certain closure properties of the class of groups which satisfy
the conjecture. We use these results to construct examples for groups which
satisfy the conjecture, and whose cohomology ring is a polynomial ring.

The second chapter deals with general
*G* modules, and not necessarily projective ones. We ask how far general
modules are from being projective. In case the group *G* is finite and we
consider *G* modules over a field *k*, the non projectivity of a
module *M* can be measured by a numerical invariant, its complexity *c _{G}(M)*.
The Alperin Evens Theorem states that the complexity can be calculated by
restricting to elementary abelian subgroups. That is-

The third chapter deals with the
restriction map from a group *G* to a finite index subgroup *H*. In
case *G* is finite, it is known by a theorem of Quillen that the kernel of
the restriction map from the cohomology ring of *G* to that of *H* is
finitely generated. We ask if this finiteness condition is true also for
infinite groups. We show that this is true if we also assume a strong
finiteness assumption on *G*, and we give a general way to construct
counterexamples when this assumption does not hold.