Question: Does anyone know groups which admit presentations with two more generators than relators and are not residually finite? If so, do we know anything about the finite residual of such groups?

Any examples of finitely presented or finitely generated groups which are not residually finite but whose inner automorphism group is residually finite?

Thanks!

Answer: 
P. Deligne has constructed non-residually finite central extensions of some arithmetic groups, so for these the inner automorphism group is residually finite, see: P. Deligne. Extensions centrales non r ́esiduellement finies de groupes arithm ́etiques. CR Acad. Sci. Paris, s ́erie A-B, 287, 203–208, 1978.

See also the central extensions constructed by D. Toledo in "Projective varieties with non-residually finite fundamental group", PUBLICATIONS MATHÉMATIQUES DE L'IHÉS, Volume 77, Number 1, 103-119: these are fundamental groups of compact manifolds, hence clearly finitely presented.
In [this](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=MR&pg8=ET&r=1&review_format=html&s4=erschler&s5=residually&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) paper, Anna Erschler constructs uncountably many non residually finite central extensions of the first Grigorchuk Group.