Loading…
FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES
Considering a certain construction of algebraic varieties X endowed with an algebraic action of the group Aut( F n ), n < ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family F of X s such that Aut( F n ) embeds into Aut( X ). For n ≥ 3, this implies nonlinear...
Saved in:
| Published in: | Transformation groups 2023-09, Vol.28 (3), p.1277-1297 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| cited_by | |
|---|---|
| cites | cdi_FETCH-LOGICAL-c270t-48bd6a888c77d7287a52c29efba85b361a1399c220ab0feaab005ebd0bfab8233 |
| container_end_page | 1297 |
| container_issue | 3 |
| container_start_page | 1277 |
| container_title | Transformation groups |
| container_volume | 28 |
| creator | POPOV, VLADIMIR L. |
| description | Considering a certain construction of algebraic varieties
X
endowed with an algebraic action of the group Aut(
F
n
),
n
< ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family
F
of
X
s such that Aut(
F
n
) embeds into Aut(
X
). For
n
≥ 3, this implies nonlinearity, and for
n
≥ 2, the existence of
F
2
in Aut(
X
) (hence nonamenability of the latter) for
X
∈
F
. We find in
F
two infinite subfamilies
N
and
R
consisting of irreducible affine varieties such that every
X
∈
N
is nonrational (and even not stably rational), while every
X
∈
F
is rational and 3
n
-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties
Z
, for which Aut(
Z
) contains the braid group
B
n
on
n
strands, does not exceed 3
n
. This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of
B
n
was proved (this latter bound is quadratic in
n
). The same upper bound also holds for Aut(
F
n
). In particular, it shows that the minimal rank of the Cremona groups containing Aut(
F
n
), does not exceed 3
n
, and the same is true for
B
n
. |
| doi_str_mv | 10.1007/s00031-023-09819-y |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2856980474</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2856980474</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-48bd6a888c77d7287a52c29efba85b361a1399c220ab0feaab005ebd0bfab8233</originalsourceid><addsrcrecordid>eNp9kE9PgzAYxhujiXP6BTyReEbfttCWIxLYSNhYGBhvTcvAuOg22-2wb786jN68vP_yPM-b_BC6x_CIAfiTBQCKfSDUh0jgyD9eoBEO3SkU7PXSzSCoH1BGrtGNtWsAzBljIzTL4ryeZk3hxUmdl_OlV2Ze3NTlrKwW03w58yZV2SzO56xK09917sXFJH2u4jzxXuIqT-s8Xd6iq1592O7up49Rk6V1MvWLcpInceG3hMPeD4ReMSWEaDlfcSK4CklLoq7XSoSaMqwwjaKWEFAa-k65CmGnV6B7pQWhdIwehtyd2X4dOruX6-3BbNxLSUTIIgEBD5yKDKrWbK01XS935v1TmaPEIL-xyQGbdNjkGZs8OhMdTNaJN2-d-Yv-x3UCy-BppA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2856980474</pqid></control><display><type>article</type><title>FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES</title><source>Springer Nature</source><creator>POPOV, VLADIMIR L.</creator><creatorcontrib>POPOV, VLADIMIR L.</creatorcontrib><description>Considering a certain construction of algebraic varieties
X
endowed with an algebraic action of the group Aut(
F
n
),
n
< ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family
F
of
X
s such that Aut(
F
n
) embeds into Aut(
X
). For
n
≥ 3, this implies nonlinearity, and for
n
≥ 2, the existence of
F
2
in Aut(
X
) (hence nonamenability of the latter) for
X
∈
F
. We find in
F
two infinite subfamilies
N
and
R
consisting of irreducible affine varieties such that every
X
∈
N
is nonrational (and even not stably rational), while every
X
∈
F
is rational and 3
n
-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties
Z
, for which Aut(
Z
) contains the braid group
B
n
on
n
strands, does not exceed 3
n
. This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of
B
n
was proved (this latter bound is quadratic in
n
). The same upper bound also holds for Aut(
F
n
). In particular, it shows that the minimal rank of the Cremona groups containing Aut(
F
n
), does not exceed 3
n
, and the same is true for
B
n
.</description><identifier>ISSN: 1083-4362</identifier><identifier>EISSN: 1531-586X</identifier><identifier>DOI: 10.1007/s00031-023-09819-y</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Automorphisms ; Braid theory ; Group theory ; Lie Groups ; Mathematics ; Mathematics and Statistics ; Topological Groups ; Upper bounds</subject><ispartof>Transformation groups, 2023-09, Vol.28 (3), p.1277-1297</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-48bd6a888c77d7287a52c29efba85b361a1399c220ab0feaab005ebd0bfab8233</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>POPOV, VLADIMIR L.</creatorcontrib><title>FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES</title><title>Transformation groups</title><addtitle>Transformation Groups</addtitle><description>Considering a certain construction of algebraic varieties
X
endowed with an algebraic action of the group Aut(
F
n
),
n
< ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family
F
of
X
s such that Aut(
F
n
) embeds into Aut(
X
). For
n
≥ 3, this implies nonlinearity, and for
n
≥ 2, the existence of
F
2
in Aut(
X
) (hence nonamenability of the latter) for
X
∈
F
. We find in
F
two infinite subfamilies
N
and
R
consisting of irreducible affine varieties such that every
X
∈
N
is nonrational (and even not stably rational), while every
X
∈
F
is rational and 3
n
-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties
Z
, for which Aut(
Z
) contains the braid group
B
n
on
n
strands, does not exceed 3
n
. This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of
B
n
was proved (this latter bound is quadratic in
n
). The same upper bound also holds for Aut(
F
n
). In particular, it shows that the minimal rank of the Cremona groups containing Aut(
F
n
), does not exceed 3
n
, and the same is true for
B
n
.</description><subject>Algebra</subject><subject>Automorphisms</subject><subject>Braid theory</subject><subject>Group theory</subject><subject>Lie Groups</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Topological Groups</subject><subject>Upper bounds</subject><issn>1083-4362</issn><issn>1531-586X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kE9PgzAYxhujiXP6BTyReEbfttCWIxLYSNhYGBhvTcvAuOg22-2wb786jN68vP_yPM-b_BC6x_CIAfiTBQCKfSDUh0jgyD9eoBEO3SkU7PXSzSCoH1BGrtGNtWsAzBljIzTL4ryeZk3hxUmdl_OlV2Ze3NTlrKwW03w58yZV2SzO56xK09917sXFJH2u4jzxXuIqT-s8Xd6iq1592O7up49Rk6V1MvWLcpInceG3hMPeD4ReMSWEaDlfcSK4CklLoq7XSoSaMqwwjaKWEFAa-k65CmGnV6B7pQWhdIwehtyd2X4dOruX6-3BbNxLSUTIIgEBD5yKDKrWbK01XS935v1TmaPEIL-xyQGbdNjkGZs8OhMdTNaJN2-d-Yv-x3UCy-BppA</recordid><startdate>20230901</startdate><enddate>20230901</enddate><creator>POPOV, VLADIMIR L.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20230901</creationdate><title>FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES</title><author>POPOV, VLADIMIR L.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-48bd6a888c77d7287a52c29efba85b361a1399c220ab0feaab005ebd0bfab8233</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Algebra</topic><topic>Automorphisms</topic><topic>Braid theory</topic><topic>Group theory</topic><topic>Lie Groups</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Topological Groups</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>POPOV, VLADIMIR L.</creatorcontrib><collection>CrossRef</collection><jtitle>Transformation groups</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>POPOV, VLADIMIR L.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES</atitle><jtitle>Transformation groups</jtitle><stitle>Transformation Groups</stitle><date>2023-09-01</date><risdate>2023</risdate><volume>28</volume><issue>3</issue><spage>1277</spage><epage>1297</epage><pages>1277-1297</pages><issn>1083-4362</issn><eissn>1531-586X</eissn><abstract>Considering a certain construction of algebraic varieties
X
endowed with an algebraic action of the group Aut(
F
n
),
n
< ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family
F
of
X
s such that Aut(
F
n
) embeds into Aut(
X
). For
n
≥ 3, this implies nonlinearity, and for
n
≥ 2, the existence of
F
2
in Aut(
X
) (hence nonamenability of the latter) for
X
∈
F
. We find in
F
two infinite subfamilies
N
and
R
consisting of irreducible affine varieties such that every
X
∈
N
is nonrational (and even not stably rational), while every
X
∈
F
is rational and 3
n
-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties
Z
, for which Aut(
Z
) contains the braid group
B
n
on
n
strands, does not exceed 3
n
. This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of
B
n
was proved (this latter bound is quadratic in
n
). The same upper bound also holds for Aut(
F
n
). In particular, it shows that the minimal rank of the Cremona groups containing Aut(
F
n
), does not exceed 3
n
, and the same is true for
B
n
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00031-023-09819-y</doi><tpages>21</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1083-4362 |
| ispartof | Transformation groups, 2023-09, Vol.28 (3), p.1277-1297 |
| issn | 1083-4362 1531-586X |
| language | eng |
| recordid | cdi_proquest_journals_2856980474 |
| source | Springer Nature |
| subjects | Algebra Automorphisms Braid theory Group theory Lie Groups Mathematics Mathematics and Statistics Topological Groups Upper bounds |
| title | FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES |
| url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-19T16%3A29%3A23IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=FAITHFUL%20ACTIONS%20OF%20AUTOMORPHISM%20GROUPS%20OF%20FREE%20GROUPS%20ON%20ALGEBRAIC%20VARIETIES&rft.jtitle=Transformation%20groups&rft.au=POPOV,%20VLADIMIR%20L.&rft.date=2023-09-01&rft.volume=28&rft.issue=3&rft.spage=1277&rft.epage=1297&rft.pages=1277-1297&rft.issn=1083-4362&rft.eissn=1531-586X&rft_id=info:doi/10.1007/s00031-023-09819-y&rft_dat=%3Cproquest_cross%3E2856980474%3C/proquest_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c270t-48bd6a888c77d7287a52c29efba85b361a1399c220ab0feaab005ebd0bfab8233%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2856980474&rft_id=info:pmid/&rfr_iscdi=true |