Loading…
Local spectral gap in simple Lie groups and applications
We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action Γ ↷ G , whenever Γ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G . This ext...
Saved in:
| Published in: | Inventiones mathematicae 2017-06, Vol.208 (3), p.715-802 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| cited_by | cdi_FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3 |
| container_end_page | 802 |
| container_issue | 3 |
| container_start_page | 715 |
| container_title | Inventiones mathematicae |
| container_volume | 208 |
| creator | Boutonnet, Rémi Ioana, Adrian Golsefidy, Alireza Salehi |
| description | We introduce a novel notion of
local spectral gap
for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action
Γ
↷
G
, whenever
Γ
is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group
G
. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83–121,
2008
; J Eur Math Soc (JEMS) 14:1455–1511,
2012
), and Benoist and de Saxcé (Invent Math 205:337–361,
2016
). We present several applications to the Banach–Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on
G
. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique
Γ
-invariant finitely additive measure defined on all bounded measurable subsets of
G
. |
| doi_str_mv | 10.1007/s00222-016-0699-8 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_hal_p</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_01449823v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2415572387</sourcerecordid><originalsourceid>FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3</originalsourceid><addsrcrecordid>eNp1kEFLxDAQhYMouK7-AG8FTx6iM0naJMdlUVcoeNFzyLbp2qXbxqQV_PdmqejJ0wwz33vMPEKuEe4QQN5HAMYYBSwoFFpTdUIWKDijyLQ8JYu0Bqo1wjm5iHEPkJaSLYgqh8p2WfSuGkNqdtZnbZ_F9uA7l5Wty3ZhmHzMbF9n1vuurezYDn28JGeN7aK7-qlL8vb48Lre0PLl6Xm9KmnFNR8pq6SEqi5gu21sgVbWdZ2jzkHxrchdI5ErLRpZa6cVFAKkRYlaIFiRzrd8SW5n33fbGR_agw1fZrCt2axKc5ylT4RWjH9iYm9m1ofhY3JxNPthCn06zzCBeS4ZVzJROFNVGGIMrvm1RTDHMM0cZnIuzDFMo5KGzZqY2H7nwp_z_6Jv7qNz6A</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2415572387</pqid></control><display><type>article</type><title>Local spectral gap in simple Lie groups and applications</title><source>Springer Nature</source><creator>Boutonnet, Rémi ; Ioana, Adrian ; Golsefidy, Alireza Salehi</creator><creatorcontrib>Boutonnet, Rémi ; Ioana, Adrian ; Golsefidy, Alireza Salehi</creatorcontrib><description>We introduce a novel notion of
local spectral gap
for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action
Γ
↷
G
, whenever
Γ
is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group
G
. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83–121,
2008
; J Eur Math Soc (JEMS) 14:1455–1511,
2012
), and Benoist and de Saxcé (Invent Math 205:337–361,
2016
). We present several applications to the Banach–Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on
G
. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique
Γ
-invariant finitely additive measure defined on all bounded measurable subsets of
G
.</description><identifier>ISSN: 0020-9910</identifier><identifier>EISSN: 1432-1297</identifier><identifier>DOI: 10.1007/s00222-016-0699-8</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Dynamical Systems ; Expanders ; Group Theory ; Lie groups ; Mathematics ; Mathematics and Statistics ; Random walk ; Set theory ; Spectra ; Subgroups</subject><ispartof>Inventiones mathematicae, 2017-06, Vol.208 (3), p.715-802</ispartof><rights>Springer-Verlag Berlin Heidelberg 2016</rights><rights>Springer-Verlag Berlin Heidelberg 2016.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3</citedby><cites>FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,776,780,881,27903,27904</link.rule.ids><backlink>$$Uhttps://hal.science/hal-01449823$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Boutonnet, Rémi</creatorcontrib><creatorcontrib>Ioana, Adrian</creatorcontrib><creatorcontrib>Golsefidy, Alireza Salehi</creatorcontrib><title>Local spectral gap in simple Lie groups and applications</title><title>Inventiones mathematicae</title><addtitle>Invent. math</addtitle><description>We introduce a novel notion of
local spectral gap
for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action
Γ
↷
G
, whenever
Γ
is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group
G
. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83–121,
2008
; J Eur Math Soc (JEMS) 14:1455–1511,
2012
), and Benoist and de Saxcé (Invent Math 205:337–361,
2016
). We present several applications to the Banach–Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on
G
. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique
Γ
-invariant finitely additive measure defined on all bounded measurable subsets of
G
.</description><subject>Dynamical Systems</subject><subject>Expanders</subject><subject>Group Theory</subject><subject>Lie groups</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Random walk</subject><subject>Set theory</subject><subject>Spectra</subject><subject>Subgroups</subject><issn>0020-9910</issn><issn>1432-1297</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kEFLxDAQhYMouK7-AG8FTx6iM0naJMdlUVcoeNFzyLbp2qXbxqQV_PdmqejJ0wwz33vMPEKuEe4QQN5HAMYYBSwoFFpTdUIWKDijyLQ8JYu0Bqo1wjm5iHEPkJaSLYgqh8p2WfSuGkNqdtZnbZ_F9uA7l5Wty3ZhmHzMbF9n1vuurezYDn28JGeN7aK7-qlL8vb48Lre0PLl6Xm9KmnFNR8pq6SEqi5gu21sgVbWdZ2jzkHxrchdI5ErLRpZa6cVFAKkRYlaIFiRzrd8SW5n33fbGR_agw1fZrCt2axKc5ylT4RWjH9iYm9m1ofhY3JxNPthCn06zzCBeS4ZVzJROFNVGGIMrvm1RTDHMM0cZnIuzDFMo5KGzZqY2H7nwp_z_6Jv7qNz6A</recordid><startdate>20170601</startdate><enddate>20170601</enddate><creator>Boutonnet, Rémi</creator><creator>Ioana, Adrian</creator><creator>Golsefidy, Alireza Salehi</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><scope>1XC</scope><scope>VOOES</scope></search><sort><creationdate>20170601</creationdate><title>Local spectral gap in simple Lie groups and applications</title><author>Boutonnet, Rémi ; Ioana, Adrian ; Golsefidy, Alireza Salehi</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Dynamical Systems</topic><topic>Expanders</topic><topic>Group Theory</topic><topic>Lie groups</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Random walk</topic><topic>Set theory</topic><topic>Spectra</topic><topic>Subgroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Boutonnet, Rémi</creatorcontrib><creatorcontrib>Ioana, Adrian</creatorcontrib><creatorcontrib>Golsefidy, Alireza Salehi</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Inventiones mathematicae</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Boutonnet, Rémi</au><au>Ioana, Adrian</au><au>Golsefidy, Alireza Salehi</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Local spectral gap in simple Lie groups and applications</atitle><jtitle>Inventiones mathematicae</jtitle><stitle>Invent. math</stitle><date>2017-06-01</date><risdate>2017</risdate><volume>208</volume><issue>3</issue><spage>715</spage><epage>802</epage><pages>715-802</pages><issn>0020-9910</issn><eissn>1432-1297</eissn><abstract>We introduce a novel notion of
local spectral gap
for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action
Γ
↷
G
, whenever
Γ
is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group
G
. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83–121,
2008
; J Eur Math Soc (JEMS) 14:1455–1511,
2012
), and Benoist and de Saxcé (Invent Math 205:337–361,
2016
). We present several applications to the Banach–Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on
G
. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique
Γ
-invariant finitely additive measure defined on all bounded measurable subsets of
G
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00222-016-0699-8</doi><tpages>88</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0020-9910 |
| ispartof | Inventiones mathematicae, 2017-06, Vol.208 (3), p.715-802 |
| issn | 0020-9910 1432-1297 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_01449823v1 |
| source | Springer Nature |
| subjects | Dynamical Systems Expanders Group Theory Lie groups Mathematics Mathematics and Statistics Random walk Set theory Spectra Subgroups |
| title | Local spectral gap in simple Lie groups and applications |
| url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-25T18%3A24%3A17IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_hal_p&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Local%20spectral%20gap%20in%20simple%20Lie%20groups%20and%20applications&rft.jtitle=Inventiones%20mathematicae&rft.au=Boutonnet,%20R%C3%A9mi&rft.date=2017-06-01&rft.volume=208&rft.issue=3&rft.spage=715&rft.epage=802&rft.pages=715-802&rft.issn=0020-9910&rft.eissn=1432-1297&rft_id=info:doi/10.1007/s00222-016-0699-8&rft_dat=%3Cproquest_hal_p%3E2415572387%3C/proquest_hal_p%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c393t-2c770cd60bbfa61a7ddd5195083b45ef713894f7d9e9806407a1719410a4432a3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2415572387&rft_id=info:pmid/&rfr_iscdi=true |