Discreteness properties of translation numbers in solvable groups.

*(English)*Zbl 0956.20039Let \(G\) be a group with a metric \(d\) which is invariant under left multiplication by \(G\), let \(\|\;\|\colon G\to\mathbb{Z}\) be defined by \(\|x\|=d(x,1_G)\) and let \(\tau(x)=\limsup_{n\to\infty}\tfrac{\|x^n\|}{n}\). This quantity is called the translation number of \(x\). A group is called translation proper if it carries a left-invariant metric in which the translation numbers of the non-torsion elements are non-zero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension \(m\) is a finite extension of \(\mathbb{Z}^m\). The author also gives two examples – one of a polycyclic group which is translation proper but not translation discrete, and another of a non-Abelian solvable group of infinite cohomological dimension which is translation discrete.

Reviewer: Alexandr Ivanov Budkin (Barnaul)

##### MSC:

20F65 | Geometric group theory |

20F16 | Solvable groups, supersolvable groups |

57M07 | Topological methods in group theory |

##### Keywords:

translation numbers; translation proper groups; left-invariant metrics; translation discrete groups; virtual cohomological dimensions; metabelian-by-finite groups; solvable groups; polycyclic groups
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