Satisfaction of existential theories in finitely presented groups and some embedding theorems

Abstract. The main result is that for every recursively enumerable existential consistent theory (in the usual language of group theory), there exists a finitely presented SQ-universal group H such that is satisfied in every nontrivial quotient of H. Furthermore if is satisfied in some group with a soluble word problem, then H can be taken with a soluble word problem. We characterize the finitely generated groups with soluble word problem as the finitely generated groups G for which there exists a finitely presented group H all of the nontrivial quotients of which embed G. We prove also that for every countable group G, there exists a 2-finitely generated SQ-universal group H such that every nontrivial quotient of H embeds G.

• Type: Journal
• Year: 2006
• Where: APAL
• Authors: Abderezak Ould Houcine
• Keywords: APAL 2006, Group, Nontrivial Quotient, Soluble Word Problem