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?
 A: 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.
