Solutions to Two Open Problems in Geometric Group Theory.

Convex Hull;Coset Growth;CAT(0);Polycyclic;Mathematics;Science;Mathematics

Sahattchieve, Jordan A.Debacker, Stephen M. ;

University of Michigan

关键词: Convex Hull;    Coset Growth;    CAT(0);    Polycyclic;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/93948/jantonov_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that with the usual action of F_m x Z^n on the metric product of a tree with R^n, every quasiconvex subgroup of F_m x Z^n is convex.This answers the question whether a quasiconvex subgroup of a CAT(0) group is a CAT(0) group in the affirmative for the groups F_m x Z^n.We also prove bounded packing in a special class of polycyclic groups, and we introduce the notion of coset growth and provide a bound for the coset growth of uniform lattices in Sol.