# Tits alternatives for groups of small cohomological dimension

## Author

**Jonathan A. Hillman**

## Status

Research Report 96-39

Date: 28 November 1996

## Abstract

A well-known theorem of Tits asserts that every finitely generated linear group
is either virtually solvable or contains a nonabelian free subgroup.
In this note we shall show that similar "Tits alternatives" hold for groups of
cohomological dimension two, 3-dimensional Poincaré duality groups and
2-knot groups. (Our arguments require some additional hypotheses.)
We also give several equivalent characterizations of coherent elementary amenable
2-knot groups, and determine the deficiencies of such groups in most cases.

## Key phrases

coherent. deficiency. 4-manifold. L2-Betti number. minimal Seifert hypersurface.
Poincaré duality group. Tits alternative.

## AMS Subject Classification (1991)

Primary: 57Q45

Secondary: 20J05, 57N13

## Content

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Sydney Mathematics and Statistics