Cone types and geodesic languages for lamplighter groups and Thompson's group F

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• 作者：Sean Cleary ; Murray Elder and Jennifer Taback
• 关键词：Regular language ; Rational growth ; Cone type ; Context-free grammar ; Counter automata ; Lamplighter groups ; Thompson's group F
• 刊名：Journal of Algebra
• 出版年：2006
• 期刊代码：69_00218693
• 类别：mt
• 出版时间：15 September 2006
• 卷：303
• 期：2
• 页码：476-500
• 文件大小：528 K

摘要

We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups L_n have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and that there is no regular language of geodesics. We show that the existence of families of “seesaw” elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.