  In General Relativity, cosmic strings are well known to produce a static,
locally flat spacetime with a wedge removed. If the tension exceeds a critical
value, the deficit angle becomes larger than $ 2\pi $, leading to a compact
exterior that ends in a conical singularity. In this work, we investigate
dynamical solutions for cosmic strings with super-critical tensions. To this
end, we model the string as a cylindrical shell of finite and stabilized
transverse width and show that there is a marginally super-critical regime in
which the stabilization can be achieved by physically reasonable matter.
  We show numerically that the static deficit angle solution is unstable for
super-critical string tensions. Instead, the geometry starts expanding in axial
direction at an asymptotically constant rate, and a horizon is formed in the
exterior, which has the shape of a growing cigar. We are able to find the
analytic form of the attractor solution describing the interior of the cosmic
string. In particular, this enables us to analytically derive the relation
between the string tension and the axial expansion rate. Furthermore, we show
that the exterior conical singularity can be avoided for dynamical solutions.
  Our results might be relevant for theories with two extra dimensions,
modeling our universe as a cosmic string with a three-dimensional axis. We
derive the corresponding Friedmann equation, relating the on-brane Hubble
parameter to the string tension or, equivalently, brane cosmological constant.
