  Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes
that can be constructed from centrally symmetric triangulations of even-sided
polygons. In this article we introduce an infinite-dimensional analogue and
prove that the group of symmetries of our construction is a semidirect product
of a degree 2 central extension of Thompson's infinite finitely presented
simple group T with the cyclic group of order 2. These results are inspired by
a similar recent analysis by the first author of the automorphism group of an
infinite-dimensional associahedron.
