  Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite
but of finite rank), u,v be any two elements in W, and i be a reduced word (of
length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i
a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove
among other things that the subgroups corresponding to different reduced words
for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also
generalize the enumeration result of the first three authors (see AG/9802093)
by showing that, under certain assumptions on u and v, the number of
Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of
simple reflections that appear in a reduced decomposition for u or v and F_2 is
the two-element field.
