BNSR-Invariants of Surface Houghton Groups

2024/03/07 by Torgerson, Noah, West, Jeremy
#20F65 #57K20 #57M07 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.04941

Abstract

The surface Houghton groups H_n are a family of groups generalizing Houghton groups H_n, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants Σ^m(PH_n) of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of H_n and PH_n having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian.

Related

• Surface Houghton groups
• Isomorphisms and commensurability of surface Houghton groups
• The Σ-invariants of Thompson's group F via Morse theory
• FPn properties of generalized Houghton groups
• Twisted Conjugacy in Houghton's groups