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Is there a one-relator group with some finite non-solvable quotient, that does not have all large alternating groups as finite quotients?

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    $\begingroup$ The Baumslag- Solitar groups are examples of one-relator groups with only solvable finite quotients. The groups $\endgroup$
    – moshe newman
    Commented Jul 7, 2014 at 16:40
  • $\begingroup$ G = < a,b | aa=bbb> and G = <a,b,c,d | abcd = dcba> have the alternating groups Alt(n) as quotients for all sufficiently large n. Is there something 'in between'? $\endgroup$
    – moshe newman
    Commented Jul 7, 2014 at 16:43
  • $\begingroup$ Moshe: I think it's only true for Baumslag-Solitar $BS(m,n)$ with $m,n$ coprime. $\endgroup$
    – YCor
    Commented Jul 7, 2014 at 16:49
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    $\begingroup$ Correct. If the powers are both divisible by p, say p>2, then you can take b to be of order p and the relation is trivially satisfied, so you can get pretty much anything as a quotient. $\endgroup$
    – moshe newman
    Commented Jul 7, 2014 at 17:21
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    $\begingroup$ Specifically, I'm trying the case aab=bba, which has some non-solvable quotients, but it does not seem to be known if it has all Alt(n) as quotients. $\endgroup$
    – moshe newman
    Commented Jul 8, 2014 at 20:12

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