Groups, Geometry, and Dynamics

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Volume 1, Issue 1, 2007, pp. 47–59free sample!
DOI: 10.4171/GGD/3

Automatically presented groups

Anna Erschler[1]

(1) Département de Mathématiques et Applications, École Normale Supérieure, 45, Rue d’Ulm, 75005, PARIS, FRANCE

We introduce the notions of automatically presented groups and piecewise automatically presented groups. We show that if G is a piecewise automatically presented group satisfying the property T of Kazhdan, then G is finite. We prove that if G is amenable and finitely presented, then G is virtually abelian. We give further restrictions for a group to be piecewise automatically presented and study properties of such groups. We also give examples of automatically presented groups.

Keywords: Property T, amenability, polycyclic group, nilpotent group, Grigorchuk topology, Cayley topology, groups generated by finite state automata, word problem, Haagerup property

Erschler Anna: Automatically presented groups. Groups Geom. Dyn. 1 (2007), 47-59. doi: 10.4171/GGD/3