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COMPLETE EMBEDDINGS OF GROUPS
Part of
- Special aspects of infinite or finite groups
- Structure and classification of infinite or finite groups
MARTIN R. BRIDSON, HAMISH SHORT
Journal:

Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024

Published online by Cambridge University Press:

26 January 2024, pp. 136-144

Print publication:

August 2024
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Every countable group G can be embedded in a finitely generated group $G^*$ that is hopfian and complete, that is, $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of G. If G has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.