  This note serves as a short and reader-friendly introduction to twisted
Brin-Thompson groups, which were recently constructed by Belk and the author to
provide a family of simple groups with a variety of interesting properties.
Most notably, twisted Brin-Thompson groups can be used to show that every
finitely generated group quasi-isometrically embeds as a subgroup of a finitely
generated simple group. Another important application is a concrete
construction of a family of simple groups with arbitrary finiteness length. In
addition to giving a concise introduction to the groups and these applications,
we also prove here a strengthening of one of the results from the original
paper. Namely, we prove that any finitely presented group acting faithfully and
oligomorphically on a set, with finitely generated stabilizers of finite
subsets, embeds in a finitely presented simple group. We believe this could
potentially lead to future progress on resolving the Boone-Higman Conjecture
for certain groups.
