A group is LEF (locally embeddable in the class of finite groups) if it embeds into an ultraproduct of finite groups. Residually finite groups are LEF and finitely presented LEF groups are residually finite. So if a finitely presented group is not residually finite, then it is not LEF either.

Most of the literature seems to be concerned with positive results. All non-examples I could find use the criterion above, in particular they are all finitely presented. What are some examples of finitely generated, infinitely presented groups that are not LEF? Among these, are there examples that have no LEF quotient? And if not, can the largest LEF quotient be determined explicitly?

iffthere is a finitely presented group $P$, a finite subset $S\subset P$ such that every homomorphism from $P$ to any finite group is non-injective, and a homomorphism $P\to G$ that is injective on $S$. $\endgroup$ – YCor Feb 23 at 9:151more comments