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Is there an infinite finitely presented group with NO nontrivial normal subgroup of finite index?

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migrated from mathoverflow.net Feb 6 '18 at 20:26

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  • $\begingroup$ See mathoverflow.net/questions/87347/the-higman-group $\endgroup$ – Mustafa Gokhan Benli Feb 6 '18 at 13:48
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    $\begingroup$ Higman's example was the first constructed; it's far from simple. Since then (starting from R. Thompson's groups $T,V$ in thee 60s) many infinite finitely presented simple groups have been constructed. $\endgroup$ – YCor Feb 6 '18 at 14:17
  • $\begingroup$ @YCor Initially I misinterpreted your first sentence! But I see now that you mean that it is far from simple in the technical sense rather than being far from easy. $\endgroup$ – Derek Holt Feb 6 '18 at 14:44
  • $\begingroup$ Yes I meant it has many normal subgroups (the maximal possible for a countable group: $2^{\aleph_0}$ normal subgroups, and is SQ-universal which is even stronger). Whether it's simpler (in the casual sense) than the Thompson groups depends on the point of view (if we want to start from a presentation, from a dynamical interpretation, what we want to do with it...). To prove from its presentation that it's both nontrivial and has no nontrivial finite quotient is tricky but reasonably elementary. $\endgroup$ – YCor Feb 6 '18 at 15:27
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Say that a group is "aperiodic" if it has no nontrivial finite quotient. This is better known as "minimally almost periodic" although this terminology is scarcely used for discrete groups. Of course, for an aperiodic group, "infinite" and "nontrivial" are equivalent conditions.

Higman constructed the first known finitely presented nontrivial aperiodic group, with presentation with generators $x_i$, $i\in\mathbf{Z}/4\mathbf{Z}$, and relators $x_ix_{i+1}x_i^{-1}=x_{i+1}^2$, $i\in\mathbf{Z}/4\mathbf{Z}$. It is now known as Higman group.

R. Thompson constructed two groups $T\subset V$ that are infinite, finitely presented and simple (as well as a subgroup of $T$ called $F$ and now best known, although Thompson was primarily interested in $V$). Since infinite simple groups are aperiodic, this yields examples, as well as all the subsequent constructions of infinite finitely presented simple groups (generalizatons by Higman called Thompson-Higman groups, Burger-Mozes groups, Kac-Moody groups thanks to Caprace-Rémy, and others).

Also, A. Olshanskii proved that every non-elementary hyperbolic group has a nontrivial finitely presented aperiodic quotient.

In spite of all this, it's not so easy to produce such groups, especially in the presence of additional requirements.

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