Factorizations of the Thompson-higman Groups, and Circuit Complexity

We consider the subgroup lpGk,1 of length preserving elements of the Thompson-Higman group Gk,1 and we show that all elements of Gk,1 have a unique lpGk,1 · Fk,1 factorization. This applies to the Thompson-Higman group Tk,1 as well. We show that lpGk,1 is a “diagonal” direct limit of finite symmetric groups, and that lpTk,1 is a k ∞ Prüfer group. We find an infinite generating set of lpGk,1 which is related to reversible boolean circuits. We further investigate connections between the Thompson-Higman groups, circuits, and complexity. We show that elements of Fk,1 cannot be one-way functions. We show that describing an element of Gk,1 by a generalized bijective circuit is equivalent to describing the element by a word over a certain infinite generating set of Gk,1; word length over these generators is equivalent to generalized bijective circuit size. We give some coNP-completeness results for Gk,1 (e.g., the word problem when elements are given by circuits), and #P-completeness results (e.g., finding the lpGk,1 · Fk,1 factorization of an element of Gk,1 given by a circuit).