Examples of Non-hyperbolic Groups
- The free rank 2 abelian group Z2 is not hyperbolic.
- More generally, any group which contains Z2 as a subgroup is not hyperbolic. In particular, lattices in higher rank semisimple Lie groups and the fundamental groups π1(S3−K) of nontrivial knot complements fall into this category and therefore are not hyperbolic.
- Baumslag–Solitar groups B(m,n) and any group that contains a subgroup isomorphic to some B(m,n) fail to be hyperbolic (since B(1,1) = Z2, this generalizes the previous example).
- A non-uniform lattice in rank 1 semisimple Lie groups is hyperbolic if and only if the associated symmetric space is the hyperbolic plane.
Read more about this topic: Hyperbolic Group
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