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Cattaneo Alberto - One of the best experts on this subject based on the ideXlab platform.
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Automorphismes non-symplectiques des variétés symplectiques holomorphes
HAL CCSD, 2018Co-Authors: Cattaneo AlbertoAbstract:We study Automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the Automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the Automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic Automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic Automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of Automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic Automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n 1.Dans la première partie de la thèse, nous classifions les Automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des Automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les Automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'Automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des Automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un Automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces Automorphismes et, en particulier, nous présentons la construction d’un nouvel Automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique
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Automorphismes non-symplectiques des variétés symplectiques holomorphes
2018Co-Authors: Cattaneo AlbertoAbstract:Nous allons étudier les Automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les Automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des Automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les Automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'Automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des Automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un Automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces Automorphismes et, en particulier, nous présentons la construction d’un nouvel Automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n 1. In the first part of the thesis we describe the Automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the Automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic Automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic Automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of Automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic Automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution
Fioravanti Elia - One of the best experts on this subject based on the ideXlab platform.
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Coarse-median preserving Automorphisms
2021Co-Authors: Fioravanti EliaAbstract:This paper has three main goals. First, we study fixed subgroups of Automorphisms of right-angled Artin and Coxeter groups. If $\varphi$ is an untwisted Automorphism of a RAAG, or an arbitrary Automorphism of a RACG, we prove that ${\rm Fix}~\varphi$ is finitely generated and undistorted. Up to replacing $\varphi$ with a power, we show that ${\rm Fix}~\varphi$ is quasi-convex with respect to the standard word metric. This implies that ${\rm Fix}~\varphi$ is separable and a special group in the sense of Haglund-Wise. By contrast, there exist "twisted" Automorphisms of RAAGs for which ${\rm Fix}~\varphi$ is undistorted but not of type $F$ (hence not special), of type $F$ but distorted, or even infinitely generated. Secondly, we introduce the notion of "coarse-median preserving" Automorphism of a coarse median group, which plays a key role in the above results. We show that Automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all Automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving. These facts also yield new or more elementary proofs of Nielsen realisation for RAAGs and RACGs. Finally, we show that, for every special group $G$ (in the sense of Haglund-Wise), every infinite-order, coarse-median preserving outer Automorphism of $G$ can be realised as a homothety of a finite-rank median space $X$ equipped with a "moderate" isometric $G$-action. This generalises the classical result, due to Paulin, that every infinite-order outer Automorphism of a hyperbolic group $H$ projectively stabilises a small $H$-tree.Comment: 70 pages, 5 figures; v3: added application to Nielsen realisation (Corollaries F and G) and reference
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Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes
'Oxford University Press (OUP)', 2021Co-Authors: Fioravanti EliaAbstract:We show that, under weak assumptions, the Automorphism group of a ${\rm CAT(0)}$ cube complex $X$ coincides with the Automorphism group of Hagen's contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger Automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more Automorphisms than the universal cover of the Davis complex.Comment: 16 pages, no figures; v2: published version, open access on IMR
Javier De La Cruz - One of the best experts on this subject based on the ideXlab platform.
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some new results on the self dual 120 60 24 code
Finite Fields and Their Applications, 2018Co-Authors: Martino Borello, Javier De La CruzAbstract:Abstract The existence of an extremal self-dual binary linear code of length 120 is a long-standing open problem. We continue the investigation of its Automorphism group, proving that Automorphisms of order 30 and 57 cannot occur. Supposing the involutions acting fixed point freely, we show that also Automorphisms of order 8 cannot occur and the Automorphism group is of order at most 120, with further restrictions. Finally, we present some necessary conditions for the existence of the code, based on shadow and design theory.
E C Turner - One of the best experts on this subject based on the ideXlab platform.
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all Automorphisms of free groups with maximal rank fixed subgroups
Mathematical Proceedings of the Cambridge Philosophical Society, 1996Co-Authors: D. J. Collins, E C TurnerAbstract:The Scott Conjecture, proven by Bestvina and Handel [ 2 ] says that an Automorphism of a free group of rank n has fixed subgroup of rank at most n . We characterise in Theorem A below those Automorphisms that realise this maximum. It follows from this characterisation, for example, that any such Automorphism has linear growth. In our paper [ 3 ], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those Automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [ 4 ].
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an Automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element
Journal of Pure and Applied Algebra, 1993Co-Authors: D. J. Collins, E C TurnerAbstract:Abstract We prove that if an Automorphism φ of a free group F of rank n has a fixed point subgroup of rank n then φ fixes a primitive element of F. With an appropriate generalisation of rank and primitivity, we prove the same statement for Automorphisms of free products.
Elia Fioravanti - One of the best experts on this subject based on the ideXlab platform.
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Automorphisms of contact graphs of rm cat 0 cube complexes
arXiv: Geometric Topology, 2020Co-Authors: Elia FioravantiAbstract:We show that, under weak assumptions, the Automorphism group of a ${\rm CAT(0)}$ cube complex $X$ coincides with the Automorphism group of Hagen's contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger Automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more Automorphisms than the universal cover of the Davis complex.
