Ethan Berkove

The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic -theory of Bianchi groups

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic n-space IHIn with finite volume orbit space. We then apply this result to show that, for any Bianchi group F, Wh(F), Ko(Zr), and Ki(Zr) vanish for i < -1.

The mod-2 cohomology of the Bianchi groups

The Bianchi groups are a family of discrete subgroups of PSL2 (C) which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.

The integral cohomology of the Bianchi groups

We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.

The lower algebraic K-theory of Fuchsian groups

Abstract. Let

An analysis of the (coloredcubes)3 puzzle

\Gamma

A geometric approach to the lower algebraic K-theory of Fuchsian groups

be a cocompact Fuchsian group. We calculate the lower algebraic K-theory of the integral group ring

Service-Learning in a Capstone Modeling Course

{\Bbb Z} \Gamma

Algebraic K-Theory of Mapping Class Groups

and find an explicit formula for

THE MOD 2 COHOMOLOGY RINGS OF SL2 OF THE IMAGINARY QUADRATIC INTEGERS

{\rm K}_{i}({\Bbb Z}\Gamma),\,i \le 1

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

, in terms of the lower K-groups of maximal finite cyclic subgroups of