Limits of almost homogeneous spaces and their fundamental groups
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909219384459264 |
|---|---|
| author | Zamora, Sergio |
| author_facet | Zamora, Sergio |
| contents | We say that a sequence of proper geodesic spaces $X_n$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_n \leq \text{Iso}(X_n)$ with $\text{diam} (X_n/G_n)\to 0$ as $n \to \infty$.
We show that if a sequence $(X_n,p_n)$ of pointed almost homogeneous spaces converges in the pointed Gromov--Hausdorff sense to a space $(X,p)$, then $X$ is a nilpotent locally compact group equipped with an invariant geodesic metric.
Under the above hypotheses, we show that if $X$ is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for $n$ large enough, $π_1(X) $ is a subgroup of a quotient of $ π_1(X_n) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_01985 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Limits of almost homogeneous spaces and their fundamental groups Zamora, Sergio Metric Geometry Group Theory 51F99 (Primary) 20F65, 14M17, 57S20 (Secondary) We say that a sequence of proper geodesic spaces $X_n$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_n \leq \text{Iso}(X_n)$ with $\text{diam} (X_n/G_n)\to 0$ as $n \to \infty$. We show that if a sequence $(X_n,p_n)$ of pointed almost homogeneous spaces converges in the pointed Gromov--Hausdorff sense to a space $(X,p)$, then $X$ is a nilpotent locally compact group equipped with an invariant geodesic metric. Under the above hypotheses, we show that if $X$ is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for $n$ large enough, $π_1(X) $ is a subgroup of a quotient of $ π_1(X_n) $. |
| title | Limits of almost homogeneous spaces and their fundamental groups |
| topic | Metric Geometry Group Theory 51F99 (Primary) 20F65, 14M17, 57S20 (Secondary) |
| url | https://arxiv.org/abs/2007.01985 |