Limit trees for free group automorphisms: universality
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915071749259264 |
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| author | Mutanguha, Jean Pierre |
| author_facet | Mutanguha, Jean Pierre |
| contents | To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's dominating exponential dynamics.
As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn--Handel's recognition theorem for atoroidal outer automorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01275 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Limit trees for free group automorphisms: universality Mutanguha, Jean Pierre Group Theory 20F65, 20E05, 20E08, 20E36 To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's dominating exponential dynamics. As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn--Handel's recognition theorem for atoroidal outer automorphisms. |
| title | Limit trees for free group automorphisms: universality |
| topic | Group Theory 20F65, 20E05, 20E08, 20E36 |
| url | https://arxiv.org/abs/2210.01275 |