# Finitely presented groups with no proper subgroups of finite index

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

## migrated from mathoverflow.netFeb 6 '18 at 20:26

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• – Mustafa Gokhan Benli Feb 6 '18 at 13:48
• 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. – YCor Feb 6 '18 at 14:17
• @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. – Derek Holt Feb 6 '18 at 14:44
• 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. – YCor Feb 6 '18 at 15:27

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).