{"title":"Géométrie de l’espace, du temps et de la\u0000causalité : la voie axiomatique","authors":"J. Tits","doi":"10.2140/IIG.2018.16.245","DOIUrl":"https://doi.org/10.2140/IIG.2018.16.245","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":" 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113948604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Étude de certains espaces métriques","authors":"J. Tits","doi":"10.2140/IIG.2018.16.37","DOIUrl":"https://doi.org/10.2140/IIG.2018.16.37","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"86 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121009297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (Homogeneous compact geometries, Transform. Groups 19 (2016), 43-58 and Erratum to: Homogeneous compact geometries, Transform. Groups, to appear). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type C3 and arise from polar actions on the Cayley plane over the division algebra of real octonions. The classification obtained by Kramer and Lytchak does not contain the claim that those two exceptional geometries are simply connected, but this holds true, as proved by Schillewaert and Struyve (On exceptional homogeneous compact geometries of type C3, Groups Geome. Dyn. 11 (2017), 1377-1399). The proof by Schillewaert and Struyve is of topological nature and relies on the main result of Kramer and Lytchak. In this paper we provide a combinatorial proof of that claim, independent of Kramer and Lytchak's result.
Kramer和Lytchak(齐次紧几何，Transform)给出了具有连通面板的不可约球面型齐次紧Tits几何的一个分类，该几何上存在连续作用于该几何上的紧旗-传递自同构群。组19(2016)，43-58和勘误:齐次紧致几何，变换。组，出现)。根据他们的主要结果，所有这些几何形状，除了两个是建筑的商。这两个例外是C3型的平面几何，它们是由实数八元数的除法代数上Cayley平面上的极性作用产生的。Kramer和Lytchak得到的分类不包含这两个例外几何单连通的主张，但这是正确的，正如Schillewaert和Struyve (On exceptions齐次紧几何的C3型，Groups Geome)所证明的那样。医学进展，11(2017)，1377-1399。Schillewaert和Struyve的证明是拓扑性质的，依赖于Kramer和Lytchak的主要结果。在本文中，我们提供了一个独立于Kramer和Lytchak结果的组合证明。
{"title":"On two nonbuilding but simply connected\u0000compact Tits geometries of type C3","authors":"A. Pasini","doi":"10.2140/iig.2019.17.221","DOIUrl":"https://doi.org/10.2140/iig.2019.17.221","url":null,"abstract":"A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (Homogeneous compact geometries, Transform. Groups 19 (2016), 43-58 and Erratum to: Homogeneous compact geometries, Transform. Groups, to appear). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type C3 and arise from polar actions on the Cayley plane over the division algebra of real octonions. The classification obtained by Kramer and Lytchak does not contain the claim that those two exceptional geometries are simply connected, but this holds true, as proved by Schillewaert and Struyve (On exceptional homogeneous compact geometries of type C3, Groups Geome. Dyn. 11 (2017), 1377-1399). The proof by Schillewaert and Struyve is of topological nature and relies on the main result of Kramer and Lytchak. In this paper we provide a combinatorial proof of that claim, independent of Kramer and Lytchak's result.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"08 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125700404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An automorphism $theta$ of a spherical building $Delta$ is called textit{capped} if it satisfies the following property: if there exist both type $J_1$ and $J_2$ simplices of $Delta$ mapped onto opposite simplices by $theta$ then there exists a type $J_1cup J_2$ simplex of $Delta$ mapped onto an opposite simplex by $theta$. In previous work we showed that if $Delta$ is a thick irreducible spherical building of rank at least $3$ with no Fano plane residues then every automorphism of $Delta$ is capped. In the present work we consider the spherical buildings with Fano plane residues (the textit{small buildings}). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of "opposition diagrams" to capture the structure of these automorphisms. Moreover we provide applications to the theory of "domesticity" in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types $mathsf{F}_4$ and $mathsf{E}_6$.
{"title":"Opposition diagrams for automorphisms of small spherical buildings","authors":"J. Parkinson, H. Maldeghem","doi":"10.2140/iig.2019.17.141","DOIUrl":"https://doi.org/10.2140/iig.2019.17.141","url":null,"abstract":"An automorphism $theta$ of a spherical building $Delta$ is called textit{capped} if it satisfies the following property: if there exist both type $J_1$ and $J_2$ simplices of $Delta$ mapped onto opposite simplices by $theta$ then there exists a type $J_1cup J_2$ simplex of $Delta$ mapped onto an opposite simplex by $theta$. In previous work we showed that if $Delta$ is a thick irreducible spherical building of rank at least $3$ with no Fano plane residues then every automorphism of $Delta$ is capped. In the present work we consider the spherical buildings with Fano plane residues (the textit{small buildings}). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of \"opposition diagrams\" to capture the structure of these automorphisms. Moreover we provide applications to the theory of \"domesticity\" in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types $mathsf{F}_4$ and $mathsf{E}_6$.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124523818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Rottey and G. Van de Voorde characterized regular pseudo-ovals of PG(3n - 1, q), q = 2^h, h >1 and n prime. Here an alternative proof is given and slightly stronger results are obtained.
S. Rottey和G. Van de Voorde刻画了PG(3n - 1, q)、q = 2^h、h >1和n '的正则伪椭圆。这里给出了另一种证明，得到了稍强一些的结果。
{"title":"Regular pseudo-hyperovals and regular pseudo-ovals in even characteristic","authors":"J. Thas","doi":"10.2140/iig.2019.17.77","DOIUrl":"https://doi.org/10.2140/iig.2019.17.77","url":null,"abstract":"S. Rottey and G. Van de Voorde characterized regular pseudo-ovals of PG(3n - 1, q), q = 2^h, h >1 and n prime. Here an alternative proof is given and slightly stronger results are obtained.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"338 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116445562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags.
{"title":"On triples of ideal chambers in\u0000A2-buildings","authors":"A. Parreau","doi":"10.2140/iig.2019.17.109","DOIUrl":"https://doi.org/10.2140/iig.2019.17.109","url":null,"abstract":"We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132619007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $ell_infty$. Then the exterior splash of $pi$ is the set of $q^2+q+1$ points on $ell_infty$ that lie on an extended line of $pi$. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry $CG(3,q)$, and hyper-reguli in $PG(5,q)$. In this article we use the Bruck-Bose representation in $PG(6,q)$ to investigate the structure of $pi$, and the interaction between $pi$ and its exterior splash. In $PG(6,q)$, an exterior splash $mathbb S$ has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversals lines in the cubic extension $PG(6,q^3)$. These transversal lines are used to characterise the carriers of $mathbb S$, and to characterise the sublines of $mathbb S$.
{"title":"The exterior splash in PG(6,q) :\u0000transversals","authors":"S. G. Barwick, Wen-Ai Jackson","doi":"10.2140/IIG.2019.17.1","DOIUrl":"https://doi.org/10.2140/IIG.2019.17.1","url":null,"abstract":"Let $pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $ell_infty$. Then the exterior splash of $pi$ is the set of $q^2+q+1$ points on $ell_infty$ that lie on an extended line of $pi$. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry $CG(3,q)$, and hyper-reguli in $PG(5,q)$. In this article we use the Bruck-Bose representation in $PG(6,q)$ to investigate the structure of $pi$, and the interaction between $pi$ and its exterior splash. In $PG(6,q)$, an exterior splash $mathbb S$ has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversals lines in the cubic extension $PG(6,q^3)$. These transversal lines are used to characterise the carriers of $mathbb S$, and to characterise the sublines of $mathbb S$.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131054920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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