Random Generation of Finitely Generated Subgroups of a Free Group

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity O(n) in the RAM model and O(n2 log2 n) in the bitcost model.

• Type: Journal
• Year: 2008
• Where: IJAC
• Authors: Frédérique Bassino, Cyril Nicaud, Pascal Weil
• Keywords: Efficient Algorithm, IJAC 2008, Rank Free Group, Subgroups