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Also, Marc Lackenby has given a characterisation of Large groups with respect to "the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order". The paper is "A characterisation of large finitely presented groups", J. Algebra 287 (2005) 458–473.
Further to the Baumslag-Pride result, there is a result of Gromov and Stohr which says that if G=⟨X;r⟩ has only one more generator than relators but such that one of the relators is a proper power then G is large. Jack Button has done some work furthering this (he has a paper, from 2008, entitled "Large Groups of Deficiency 1"). But the proper power result is already pretty powerful - it gives you, for instance, that one-relator groups with torsion are Large.
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).
To prove that $\mathbb{Z}$ is not quasi-isometric to $\mathbb{Z}^2$, you could thinks about the number of ends of your Cayley graph. Ends are a quasi-isometric invariant, and quite clearly $\mathbb{Z}$ is 2-ended while $\mathbb{Z}^2$ is 1-ended. Of course, this doesn't answer your specific questions...but it does tell you why they are not quasi-isometric...
Thanks. However, it seems I have got my question upside-down. I meant to ask about residually finite-by-cyclic groups, not cyclic-by-residually finite...Sorry!
I apologise for my sloppyness - it seems I have got my question "upside-down", as it were. I meant to ask about residually finite-by-cyclic groups, not cyclic-by-residually finite! Sorry! (But thankyou for your answer, it is informative all the same.)
I've never thought of mining engineering being related to maths. Possibly the world's most famous mining engineer, Major-General Sir Richard Hannay, KCB, OBE, DSO, Legion of Honour, never struck me as the maths-ey type...
I was taking Aut(G) to be a representation for the abstract group Aut(G). I am not meaning linear representations, just representations in a more general sense (realisations?).