Rational growth in the Heisenberg group

Moon Duchin, Michael Shapiro

Published 2014-11-16Version 1

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. (In the polynomial growth range, being rational is the same as being eventually quasi-polynomial.) A long-standing open question asks whether the Heisenberg group has rational growth with respect to all finite generating sets, and we settle this question affirmatively. Previously, the only groups known to have this property were virtually abelian groups and hyperbolic groups. Our method involves a very precise description of families of geodesics (in any word metric) that suffice to represent all group elements.