Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type
Let Г be a torsion-free uniform lattice of SU( m , 1), m > 1. Let G be either SU( p , 2) with p ≥ 2, or SO( p , 2) with p ≥ 3. The symmetric spaces associated to these G ’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the corres...
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| Veröffentlicht in: | Geometriae dedicata 2008-12, Vol.137 (1), p.85-111 |
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| container_title | Geometriae dedicata |
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| creator | Koziarz, Vincent Maubon, Julien |
| description | Let Г be a torsion-free uniform lattice of SU(
m
, 1),
m
> 1. Let
G
be either SU(
p
, 2) with
p
≥ 2,
or SO(
p
, 2) with
p
≥ 3. The symmetric spaces associated to these
G
’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into
G
. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily
G
= SU(
p
, 2) with
p
≥ 2
m
and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(
p
, 2)/S(U(
p
) × U(2)), on which it acts cocompactly. |
| doi_str_mv | 10.1007/s10711-008-9288-3 |
| format | Article |
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m
, 1),
m
> 1. Let
G
be either SU(
p
, 2) with
p
≥ 2,
or SO(
p
, 2) with
p
≥ 3. The symmetric spaces associated to these
G
’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into
G
. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily
G
= SU(
p
, 2) with
p
≥ 2
m
and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(
p
, 2)/S(U(
p
) × U(2)), on which it acts cocompactly.</description><identifier>ISSN: 0046-5755</identifier><identifier>EISSN: 1572-9168</identifier><identifier>DOI: 10.1007/s10711-008-9288-3</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebraic Geometry ; Convex and Discrete Geometry ; Differential Geometry ; Hyperbolic Geometry ; Mathematics ; Mathematics and Statistics ; Original Paper ; Projective Geometry ; Topology</subject><ispartof>Geometriae dedicata, 2008-12, Vol.137 (1), p.85-111</ispartof><rights>Springer Science+Business Media B.V. 2008</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-c365t-e0c73db3000941b2feb36e19f37b6bce282c60467a87a3ce6b2c6f5a8a321d803</citedby><cites>FETCH-LOGICAL-c365t-e0c73db3000941b2feb36e19f37b6bce282c60467a87a3ce6b2c6f5a8a321d803</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10711-008-9288-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10711-008-9288-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,776,780,881,27901,27902,41464,42533,51294</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00135135$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Koziarz, Vincent</creatorcontrib><creatorcontrib>Maubon, Julien</creatorcontrib><title>Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type</title><title>Geometriae dedicata</title><addtitle>Geom Dedicata</addtitle><description>Let Г be a torsion-free uniform lattice of SU(
m
, 1),
m
> 1. Let
G
be either SU(
p
, 2) with
p
≥ 2,
or SO(
p
, 2) with
p
≥ 3. The symmetric spaces associated to these
G
’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into
G
. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily
G
= SU(
p
, 2) with
p
≥ 2
m
and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(
p
, 2)/S(U(
p
) × U(2)), on which it acts cocompactly.</description><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Hyperbolic Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><subject>Projective Geometry</subject><subject>Topology</subject><issn>0046-5755</issn><issn>1572-9168</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKs_wFuuHqKTpLvJHktRKywIoueQTbNt6nazJKnovzfrikdhYJjhfcO8h9A1hVsKIO4iBUEpAZCkYlISfoJmtBCMVLSUp2gGsChJIYriHF3EuAeASgg2Q_rFDsFG2yednO8j9i02_jB09hPvvgYbGt85gzudkjM2Ytcnj4Pu3zHDptMxOqM7XDuLt8Efhx9-bcPBJad7nPKFS3TW6i7aq98-R28P96-rNamfH59Wy5oYXhaJWDCCbxo-fragDWttw0tLq5aLpmyMZZKZMpsQWgrNjS2bPLeFlpozupHA5-hmurvTnRqCO-jwpbx2ar2s1bgDoLzI9cGylk5aE3yMwbZ_AAU15qmmPDMj1Zin4plhExOztt_aoPb-GPps6R_oG2tOePQ</recordid><startdate>20081201</startdate><enddate>20081201</enddate><creator>Koziarz, Vincent</creator><creator>Maubon, Julien</creator><general>Springer Netherlands</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope></search><sort><creationdate>20081201</creationdate><title>Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type</title><author>Koziarz, Vincent ; Maubon, Julien</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c365t-e0c73db3000941b2feb36e19f37b6bce282c60467a87a3ce6b2c6f5a8a321d803</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Hyperbolic Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><topic>Projective Geometry</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Koziarz, Vincent</creatorcontrib><creatorcontrib>Maubon, Julien</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Geometriae dedicata</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Koziarz, Vincent</au><au>Maubon, Julien</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type</atitle><jtitle>Geometriae dedicata</jtitle><stitle>Geom Dedicata</stitle><date>2008-12-01</date><risdate>2008</risdate><volume>137</volume><issue>1</issue><spage>85</spage><epage>111</epage><pages>85-111</pages><issn>0046-5755</issn><eissn>1572-9168</eissn><abstract>Let Г be a torsion-free uniform lattice of SU(
m
, 1),
m
> 1. Let
G
be either SU(
p
, 2) with
p
≥ 2,
or SO(
p
, 2) with
p
≥ 3. The symmetric spaces associated to these
G
’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into
G
. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily
G
= SU(
p
, 2) with
p
≥ 2
m
and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(
p
, 2)/S(U(
p
) × U(2)), on which it acts cocompactly.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10711-008-9288-3</doi><tpages>27</tpages><oa>free_for_read</oa></addata></record> |
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| issn | 0046-5755 1572-9168 |
| language | eng |
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| source | SpringerLink_现刊 |
| subjects | Algebraic Geometry Convex and Discrete Geometry Differential Geometry Hyperbolic Geometry Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology |
| title | Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type |
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