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ADL
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Groups surjecting onto a free group
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.
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Groups surjecting onto a free group
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.
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Nielsen equivalence in one-relator groups
@Boris Novikov: $X$ is an $n$-tuple of words in the free group $F_n$.
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Nielsen equivalence in one-relator groups
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Non-residually finite groups
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).
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Question about the proof of the fact that IR is not quasi-isomtric to IR^2
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...
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residually finite-by-$\mathbb{Z}$ groups are residually finite
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!
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residually finite-by-$\mathbb{Z}$ groups are residually finite
(I have edited the question appropriately now.)
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residually finite-by-$\mathbb{Z}$ groups are residually finite
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.)
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Famous mathematicians with background in arts/humanities/law etc
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...
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Decision problems and group representations
Ah, okay, thanks for clearing up my misconceptions!
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Decision problems and group representations
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?).
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