The rationality of Sol manifolds

Abstract

Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $\Gamma$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by enumerating $G$ by a regular language. However, in contrast to most earlier proofs of this sort our regular language is not a language of words in the generating set, but rather reflects a different geometric structure in $G$.

30 pages; author's name changed to agree with published version; to appear in Journal of Algebra

References in corpus (1)

• Automatic structures, rational growth and geometrically finite hyperbolic groups

Cited by in corpus (3)

• Negligibity of elliptic elements in ascending HNN-extensions of ${\mathbb{Z}}^m$
• Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles
• The Conjugacy Ratio of Abelian-by-Cyclic Groups