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 the 2x2 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 towards analysing fundamental decision problems in 2x2 matrix semigroups.

翻译：将词结构嵌入到矩阵半群中，为词组合学与线性代数之间提供了自然的桥梁。然而，低维矩阵半群对可能的嵌入施加了严格的结构限制。某些有限生成群允许在SL(2, C)及其他类似矩阵群中存在忠实表示。另一方面，已知两个生成元上的两个自由半群的积无法嵌入到2x2复矩阵中。本文研究了词结构在复数域上低维矩阵半群中的嵌入问题，并发展了构造欧几里得比安基群词表示的新技术。这些表示提供了一个符号框架，并构成了分析2x2矩阵半群中基本判定问题的自然第一步。