2014/01/08 by Brian Ray, Ray, Brian
Mathematics · #20F65 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20F65
paper · pdf · doi:10.48550/arxiv.1401.1862
10 pages
arxiv created 2014/01/08 · arxiv updated 2014/01/10
A subset Σ⊂ FN of the free group of rank N is called spectrally rigid if whenever trees T, T' in Culler-Vogtmann Outer Space are such that ‖ g ‖T = ‖ g ‖T' for every g ∈ Σ, it follows that T = T'. Results of Smillie, Vogtmann, Cohen, Lustig, and Steiner prove that (for N ≥ 2) no finite subset of FN is spectrally rigid in FN. We prove that if \ Hi \i=1k is a finite collection of subgroups, each of infinite index, and gi ∈ FN, then ∪i=1k gi Hi is not spectrally rigid in FN. Taking Hi = 1, we recover the results about finite sets. We also prove that any coset of a nontrivial normal subgroup H \lhd FN is spectrally rigid.