Boundedness of certain automorphism groups of an open manifold

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It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold with finite number of ends are bounded. It follows that these groups are uniformly perfect. In order to characterize the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that the commutator length diameter of the automorphism group D(M) of a portable manifold M is estimated by 2fragdD₍ₘ₎+2, where fragdD₍ₘ₎ is the diameter of D(M) in the fragmentation norm.