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, graph 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 $F$. Finally, randomized streaming algorithms for subgroup membership problems in free groups and direct products of free groups are studied.

翻译：我们研究了有限生成群的词问题的确定性与随机化流式算法。对于有限生成线性群、亚交换群和自由可解群，我们证明了其词问题存在具有对数空间复杂度的随机化流式算法。我们还表明，具有对数空间随机化流式算法处理词问题的有限生成群类在若干群论构造下封闭：有限扩张、图积以及自由交换群的圈积。我们通过若干下界结果与这些结论形成对比。一个有限呈现群的例子是汤普森群$F$，其词问题仅存在线性空间随机化流式算法。最后，研究了自由群及其直积中子群成员问题的随机化流式算法。