Conjugation-invariant norms on groups of geometric origin

Abstract

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.

On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds

“…Section 4 includes main results in the odd-dimensional case. In Section 4.1 we recall the basic procedure in [4,17] for removing crossing points on the track of an isotopy and a factorization of an isotopy. We Section 5 includes main results in the even-dimensional case.…”

“…We recall basic facts on conjugation-invariant norms [4,18]. An extended conjugation-invariant norm on G is a function q : G → [0, ∞] which satisfies the following conditions : for any g, h ∈ G…”

“…The problem of the uniform perfectness of diffeomorphism groups has been studied by Burago, Ivanov and Polterovich [4], Tsuboi [17], [19] and others. They showed that the diffeomorphism groups are uniformly perfect for all closed manifolds except for two and four dimensional cases.…”

“…In Section 2.4 we separate the notions of absorption and displacement properties for compact subsets in a manifold from the related arguments. We clarify their basic behavior and roles in the basic factorization lemma on diffeomorphisms in [4,17,18] and extend the factorization lemma to the case of manifolds with boundary.…”

Conjugacy invariant pseudo-norms, representability and RNA secondary structures

“…The formulation of RNA folding we consider is a special case of an estimate for the c-conjugating norm, as in [1]. This suggests that mathematical work in [1] and related papers may shed light on RNA folding.…”