No Access
NON-RIGIDITY OF CYCLIC AUTOMORPHIC ORBITS IN FREE GROUPS
https://doi.org/10.1142/S021819671250021XCited by:1 (Source: Crossref)
Abstract
We say a subset Σ ⊆ FN of the free group of rank N is spectrally rigid if whenever T1, T2 ∈ cvN are ℝ-trees in (unprojectivized) outer space for which ‖σ‖T1 = ‖σ‖T2 for every σ ∈ Σ, then T1 = T2 in cvN. The general theory of (non-abelian) actions of groups on ℝ-trees establishes that T ∈ cvN is uniquely determined by its translation length function ‖⋅‖T : FN → ℝ, and consequently that FN itself is spectrally rigid. Results of Smillie and Vogtmann, and of Cohen, Lustig and Steiner establish that no finite Σ is spectrally rigid. Capitalizing on their constructions, we prove that for any Φ ∈ Aut(FN) and g ∈ FN, the set Σ = {Φn(g)}n∈ℤ is not spectrally rigid.
Keywords:
AMSC: 20F65
