Solutions to two open problems in geometric group theory

2025/05/09 by Sahattchieve, Jordan A.
#20F16 #20F19 #20F65 #20F67 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2505.07873

Abstract

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 Fm x Zn on the metric product of the Cayley graph of Fm with Rn, every quasiconvex subgroup of Fm x Zn 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 Zn. 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.

Citations

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• The entropy formula for the Ricci flow and its geometric applications
• The Geometries of 3-Manifolds

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