Boundedly Generated Group - Free Groups Are Not Boundedly Generated

Free Groups Are Not Boundedly Generated

Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups.

Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them. Similarly, since SL2(Z) contains F2 as a subgroup of index 12, it is enough to consider SL2(Z). In other words, to show that no Fn with n ≥ 2 has bounded generation, it is sufficient to prove this for one of them or even just for SL2(Z) .

Read more about this topic:  Boundedly Generated Group

Other articles related to "free groups are not boundedly generated, group, free group":

Boundedly Generated Group - Free Groups Are Not Boundedly Generated - Gromov Boundary
... of the action of hyperbolic elements on the Gromov boundary of a Gromov-hyperbolic group ... For the special case of the free group Fn, the boundary (or space of ends) can be identified with the space X of semi-infinite reduced words g1 g2 ··· in the generators and ... The free group acts by left multiplication on the semi-infinite words ...

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