Examples
- If n ≥ 3, the group SLn(Z) is boundedly generated by its elementary subgroups, formed by matrices differing from the identity matrix only in one off-diagonal entry. In 1984, Carter and Keller gave an elementary proof of this result, motivated by a question in algebraic K-theory.
- A free group on at least two generators is not boundedly generated (see below).
- The group SL2(Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12.
- A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.
Read more about this topic: Boundedly Generated Group
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