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A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.Note that the norm q K has the following extremal property: for any norm q bounded on K there is a constant λ such that q ≤ λq K . Hence, if K is finite, the group G is bounded if and only if q K is bounded.Example 1.1. Groups SL(n, R) for n ≥ 2 and SL(n, Z) for n ≥ 3 are finitely c-generated by the set K of all elementary matrices whose off-diagonal term equals ±1. Moreover we claim that the number of terms in the decomposition (1) is bounded by a constant which does not depend on h.In the case of SL(n, R) the claim follows from an appropriate version of the Gauss elimination process.As for SL(n, Z), denote by E the set of all elementary matrices whose only non-zero off-diagonal element equals to 1. There exists N = N(n) ∈ N so that every element from SL(n, Z) can be written as a product of ≤ N matrices of the form E p , where E ∈ E and p ∈ Z (in other words, SL(n, Z) possesses a bounded generation by elements from E), see [26]. The claim readily follows from the fact that each E p = [A, B p ] for some A, B ∈ E. Let us prove this identity: let E ij (where i = j) denotes the elementary matrix from E whose only non-zero off-diagonal element stands in the i-th raw and j-th column. Without loss of generality, put i = 1, j = 3. Then E p 13 = [E 12 , E p 23 ] as required. It follows from the claim that the "extremal" norm q K is bounded, and hence the groups in question are bounded in view of extremality of q K .Example 1.2. The commutator length. Given a group G, denote by G ′ its commutator subgroup. The norm on G ′ c-generated by the set of all simple commutators [a, b] = aba −1 b −1 is called the commutator length and is denoted by cl G . The role of the commutator subgroupThe next observations suggest that the commutator subgroup plays a significant role in the study of boundedness.

We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.

We say that a Riemannian manifold .M; g/ with a non-empty boundary @M is a minimal orientable filling if, for every compact orientable .We show that if a metric g on a region M R n with a connected boundary is sufficiently C 2 -close to a Euclidean one, then it is a minimal filling. By studying the equality case vol.y/ for all x; y 2 @M then .M; g/ is isometric to . f M ; Q g/. This gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of Michel's conjecture.

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