Let
P
P
be the group of all the orientation preserving piecewise linear homeomorphisms of the interval
[
0
,
1
]
[0,1]
. Given any
a
>
1
a>1
, let
P
a
P^a
be the subgroup of
P
P
consisting of all the elements with slopes in
a
Z
a^\mathbb {Z}
, and let
P
Q
P^\mathbb {Q}
be the subgroup of
P
P
consisting of all the elements with slopes and breaks in
Q
\mathbb {Q}
. We show that the groups
P
P
,
P
a
P^a
,
P
Q
P^\mathbb {Q}
, as well as Thompson group
F
F
, are invariably generated.