Finite index subgroups in Chevalley groups are bounded: an addendum to "On bi-invariant word metrics"

A group G is called bounded if every conjugation invariant norm on G has finite diameter. Examples of bounded groups include SLn(Z) for n ≥ 3, Diff0(M), where M is a manifold of dimension different from
2 and 4, the commutator subgroup of Thompson’s group F and many others. A finite index subgroup of a bounded group does not have to be bounded. The simplest example is the infinite cyclic subgroup of the infinite dihedral group. The purpose of this note is to prove the following result. Theorem: Let G be a Chevalley group over the ring of S-integers in a number field k constructed from a root system whose irreducible components all have rank at least 2. If H is commensurable with G then it is bounded. The above theorem generalises the main result of the paper [7]. The proof is similar with an additional ingredient being an explicit form of bounded generation of a finite index subgroup of a boundedly generated group. We also correct a couple of mistakes from the original proof. First, the reduction to rank two sublattices in the proof of [7, Theorem 1.1] needs an extra argument if α is not contained in a rank two root subsystem isomorphic to A2. This is done in Lemma 1 of the current paper. Second, the same proof in [7] erroneously assumed that (OS, +) is always a finitely generated abelian group. Lemma 2 of the current paper fixes the resulting gap in the proof.

Motivation and context. The study of general conjugation-invariant norms have several sources [5]. In finite groups there is a well studied notion of a covering number [6]. Moreover, generation by conjugacy classes of finite simple groups has been extensively investigated [10, 11]. In symplectic geometry, there is a natural conjugation-invariant norm, called the Hofer norm [14], on the Hamiltonian transformations of a symplectic manifold. General conjugation-invariant norms can be used in understanding Hamiltonian group actions on symplectic manifolds. For example, in [9] it is shown that certain bounded groups don’t admit Hamiltonian actions on symplectic manifolds. In differential topology diffeomorphisms of manifolds have fragmentation property [2]. That is, they can be expressed as composition of diffeomorphisms supported in balls, for example. Investigation how complicated such decompositions are can be done in the framework of conjugation invariant norms [4].