#### Year of Publication

2010

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Marian Anton

#### Second Advisor

Dr. Edgar Enochs

#### Abstract

A motivational problem for group homology is a conjecture of Quillen that states, as reformulated by Anton, that the second homology of the general linear group over *R* = *Z*[1/*p*; ζ* _{p}*], for

*p*an odd prime, is isomorphic to the second homology of the group of units of

*R*, where the homology calculations are over the field of order

*p*. By considering the group extension spectral sequence applied to the short exact sequence 1 →

*SL*

_{2}→

*GL*

_{2}→

*GL*

_{1}→ 1 we show that the calculation of the homology of

*SL*

_{2}gives information about this conjecture. We also present a series of algorithms that finds an upper bound on the second homology group of a finitely-presented group. In particular, given a finitely-presented group

*G*, Hopf's formula expresses the second integral homology of

*G*in terms of generators and relators; the algorithms exploit Hopf's formula to estimate

*H*

_{2}(

*G*;

*k*), with coefficients in a finite field

*k*. We conclude with sample calculations using the algorithms.

#### Recommended Citation

Roberts, Joshua D., "ALGORITHMS FOR UPPER BOUNDS OF LOW DIMENSIONAL GROUP HOMOLOGY" (2010). *University of Kentucky Doctoral Dissertations.* Paper 104.

http://uknowledge.uky.edu/gradschool_diss/104