Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings. Certain finitely generated groups admit faithful representations in SL(2,C) and other similar matrix groups. On the other hand, it is known that the product of two free semigroups on two generators cannot be embedded into 2×2 complex matrices.
In this paper we study embeddings of word structures into low-dimensional matrix semigroups over the complex numbers and develop new techniques for constructing word representations of the Euclidean Bianchi groups. These representations provide a symbolic framework and a natural first step for analysing fundamental decision problems in 2×2 matrix semigroups.