Isometric embeddings of surfaces for scl
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916582150635520 |
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| author | Marchand, Alexis |
| author_facet | Marchand, Alexis |
| contents | Let $φ:F_1\to F_2$ be an injective morphism of free groups. If $φ$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $φ$ is an isometric embedding for stable commutator length. More generally, we show that if $T$ is a subsurface of an oriented compact (possibly closed) connected surface $S$, and $c$ is an integral $1$-chain on $π_1T$, then there is an isometric embedding $H_2(T,c)\to H_2(S,c)$ for the relative Gromov seminorm. Those statements are proved by finding an appropriate standard form for admissible surfaces and showing that, under the right homology vanishing conditions, such an admissible surface in $S$ for a chain in $T$ is in fact an admissible surface in $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04133 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isometric embeddings of surfaces for scl Marchand, Alexis Geometric Topology Group Theory 20F65, 20J05, 57M07 Let $φ:F_1\to F_2$ be an injective morphism of free groups. If $φ$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $φ$ is an isometric embedding for stable commutator length. More generally, we show that if $T$ is a subsurface of an oriented compact (possibly closed) connected surface $S$, and $c$ is an integral $1$-chain on $π_1T$, then there is an isometric embedding $H_2(T,c)\to H_2(S,c)$ for the relative Gromov seminorm. Those statements are proved by finding an appropriate standard form for admissible surfaces and showing that, under the right homology vanishing conditions, such an admissible surface in $S$ for a chain in $T$ is in fact an admissible surface in $T$. |
| title | Isometric embeddings of surfaces for scl |
| topic | Geometric Topology Group Theory 20F65, 20J05, 57M07 |
| url | https://arxiv.org/abs/2302.04133 |