We study deterministic and randomized streaming algorithms for word problems of finitely generated groups. For finitely generated linear groups, metabelian groups and free solvable groups we show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems. We also show that the class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group theoretical constructions: finite extensions, direct products, free products and wreath products by free abelian groups. We contrast these results with several lower bound. An example of a finitely presented group, where the word problem has only a linear space randomized streaming algorithm, is Thompson's group $F$.

翻译：我们研究有限生成群词汇问题的确定性与随机化流算法。对于有限生成线性群、亚贝尔群与自由可解群，我们证明了其词汇问题存在具有对数空间复杂度的随机化流算法。我们还表明，对于词汇问题存在对数空间随机化流算法的有限生成群类，在几种群论构造下保持封闭性：有限扩张、直积、自由积以及被自由阿贝尔群的圈积。我们将这些结果与若干下界进行对比。一个有限展示群的示例——汤普森群$F$的词汇问题仅存在线性空间随机化流算法。