Abstract
Let T n denote the group of real n × n upper-triangular matrices with 1s on the diagonal. This paper constructs left-invariant Riemannian and sub-Riemannian metrics on T3 ⊕ T4 whose geodesic flow has a subsystem that factors onto a suspended horseshoe. As a corollary, left-invariant Riemannian metrics with positive topological entropy are constructed on all quotients D∖T n where D is a discrete subgroup of T n and n ≥ 7.
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