Generic norms and metrics on countable abelian groups

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00504468" target="_blank" >RIV/67985840:_____/19:00504468 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s00605-019-01277-7" target="_blank" >http://dx.doi.org/10.1007/s00605-019-01277-7</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00605-019-01277-7" target="_blank" >10.1007/s00605-019-01277-7</a>

Result language
angličtina
Original language name
Generic norms and metrics on countable abelian groups
Original language description
For a countable abelian group G we investigate generic properties of the space of all invariant metrics on G. We prove that for every such an unbounded group G, i.e. group which has elements of arbitrarily high order, there is a dense set of invariant metrics on G which make G isometric to the rational Urysohn space, and a comeager set of invariant metrics such that the completion is isometric to the Urysohn space. This generalizes results of Cameron and Vershik, Niemiec, and the author. Then we prove that for every countable abelian G such that G≅ ⨁ N G there is a comeager set of invariant metrics on G such that all of them give rise to the same metric group after completion. If moreover G is unbounded, then using a result of Melleray and Tsankov we get that the completion is extremely amenable.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF16-34860L" target="_blank" >GF16-34860L: Logic and Topology in Banach spaces</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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