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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!

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    $\begingroup$ Finitely generated subgroups of residually finite groups, so just take $\langle a, b, c; b^{-1}a^2b=a^3\rangle$. This is non-residually finite, as it contains $BS(2, 3)$ which is not even Hopfian! (in fact, the above group is non-Hopfian too, as the epimorphism lifts). $\endgroup$
    – ADL
    Oct 27, 2011 at 16:39
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    $\begingroup$ Take any finitely presented non-residually finite group $G$ and any free group $F$ of rank $k$. Choosing $k$ large enough one can make sure that the free product $H=G∗F$ has arbitrary large deficiency. The finite residual of $H$ will coincide with the finite residual of $G$, over which you have almost no control. On the other hand, I do not know whether a group of deficiency $2$ can have non-trivial center... $\endgroup$ Oct 27, 2011 at 20:24
  • $\begingroup$ Thanks for the suggestions Alan and Ashot. For Ashot: I thought about it, but I can't see how the residual of $H$ coincides with that of $G$. The following, I think, says that they are not the same: Denote the residual of $G$ by $R_G$. Then the conjugate of $R_G$ by an element from the generating set of $F$ is not contained in $R_G$. Therefore, $R_G$ is not normal in $G*F$. Or did you mean that the residual of $H$ is obtained by taking the free product of $R_G$ with $F$? $\endgroup$
    – Mariano
    Oct 31, 2011 at 9:52
  • $\begingroup$ @Mariano: you are right, the finite residual of $H$ will be the normal closure of $R_G$ in $H$, which is the free product ${\large ∗}_{t\in T} (tR_Gt^{-1})$, where $T$ is some set of left coset representatives of $G$ in $H$. $\endgroup$ Oct 31, 2011 at 20:36
  • $\begingroup$ @Ashot: Sorry for the delay in this question. It was not until recently that I carefully thought about your last comment. I understand why $\ast_{t\in T(tR_Gt^{-1})}$ is contained in $R_H$, but how do we know they are the same? Can we say the same about the following more general situation?: Take $G=G_1\ast G_2$; is $R_G$ the free product $\ast_{t\in T}(tR_{G_1}t^{-1})\ast_{s\in S}(sR_{G_2}s^{-1})$, where $T$ is a set of left coset representatives for $G_1$ in $G$ and $S$ a set of left coset representatives for $G_2$ in $G$? $\endgroup$
    – Mariano
    Sep 17, 2012 at 19:21

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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.

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In this paper, Anna Erschler constructs uncountably many non residually finite central extensions of the first Grigorchuk Group.

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