  We present several algebraic and differential-geometric constructions of
tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov
tetrahedron equation. In particular, we obtain a family of new (nonlinear)
polynomial tetrahedron maps on the space of square matrices of arbitrary size,
using a matrix refactorisation equation, which does not coincide with the
standard local Yang--Baxter equation. Liouville integrability is established
for some of these maps.
  Also, we show how to derive linear tetrahedron maps as linear approximations
of nonlinear ones, using Lax representations and the differentials of nonlinear
tetrahedron maps on manifolds. We apply this construction to two nonlinear
maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton
solutions of vector KP equations and a tetrahedron map obtained in
[arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a
Darboux matrix associated with a Lax operator for the NLS equation. We derive
parametric families of new linear tetrahedron maps (with nonlinear dependence
on parameters), which are linear approximations for these nonlinear ones.
  Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron
map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix
generalisations can be regarded as tetrahedron maps in noncommutative
variables.
  Besides, several tetrahedron maps on arbitrary groups are constructed.
