  We consider the subgroup lpG_{k,1} of length preserving elements of the
Thompson-Higman group G_{k,1} and we show that all elements of G_{k,1} have a
unique lpG_{k,1}.F_{k,1} factorization. This applies to the Thompson-Higman
group T_{k,1} as well. We show that lpG_{k,1} is a ``diagonal'' direct limit of
finite symmetric groups, and that lpT_{k,1} is a k^infinity Pr"ufer group. We
find an infinite generating set of lpG_{k,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 F_{k,1} cannot be one-way
functions. We show that describing an element of G_{k,1} by a generalized
bijective circuit is equivalent to describing the element by a word over a
certain infinite generating set of G_{k,1}; word length over these generators
is equivalent to generalized bijective circuit size.
  We give some coNP-completeness results for G_{k,1} (e.g., the word problem
when elements are given by circuits), and #P-completeness results (e.g.,
finding the lpG_{k,1}.F_{k,1} factorization of an element of G_{k,1} given by a
circuit).
