Boundedly Generated Group - Properties

Properties

  • Bounded generation is unaffected by passing to a subgroup of finite index: if H is a finite index subgroup of G then G is boundedly generated if and only if H is boundedly generated.
  • Any quotient group of a boundedly generated group is also boundedly generated.
  • A finitely generated periodic group must be finite if it is boundedly generated; equivalently, an infinite finitely generated periodic group is not boundedly generated.

A pseudocharacter on a discrete group G is defined to be a real-valued function f on a G such that

f(gh) โˆ’ f(g) โˆ’ f(h) is uniformly bounded and f(gn) = nยทf(g).
  • The vector space of pseudocharacters of a boundedly generated group G is finite dimensional.

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