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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 1

### An algorithm for low dimensional group homology

Pages: 27 – 37

DOI: http://dx.doi.org/10.4310/HHA.2010.v12.n1.a3

#### Author

#### Abstract

Given a finitely presented group $G$, Hopf’s formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf’s formula to estimate $H_2(G;k)$, with coefficients in a finite field $k$, and give examples using $G=SL_2$ over specific rings of integers. These examples are related to a conjecture of Quillen.

#### Keywords

Group homology; experimental mathematics; l algebra; computationa; Hopf’s formula

#### 2010 Mathematics Subject Classification

20G30, 20J06

Published 1 January 2010

**Supplemental file:** GAP code (text)