Representations of the fundamental group and the torsor of deformations. Local study

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Galois Cohomology ◽  
Profinite Groups ◽  
Local Study ◽  
Natural Equivalence ◽  
Logarithmic Geometry ◽  
Continuous Cohomology ◽  
Affine Scheme

This chapter focuses on representations of the fundamental group and the torsor of deformations. It considers the case of an affine scheme of a particular type, qualified also as small by Faltings. It introduces the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then constructs a natural equivalence between these two categories. It proves that this approach generalizes simultaneously Faltings' construction for small generalized representations and Hyodo's theory of p-adic variations of Hodge–Tate structures. The discussion covers the relevant notation and conventions, results on continuous cohomology of profinite groups, objects with group actions, logarithmic geometry lexicon, Faltings' almost purity theorem, Faltings extension, Galois cohomology, Fontaine p-adic infinitesimal thickenings, Higgs–Tate torsors and algebras, Dolbeault representations, and small representations. The chapter also describes the descent of small representations and applications and concludes with an analysis of Hodge–Tate representations.

Representations of the fundamental group and the torsor of deformations. An overview

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Higgs Bundle ◽  
New Approach ◽  
Natural Equivalence ◽  
Affine Scheme ◽  
Logical Links

This chapter provides an overview of a new approach to the p-adic Simpson correspondence, focusing on representations of the fundamental group and the torsor of deformations. The discussion covers the notation and conventions, small generalized representations, the torsor of deformations, Faltings ringed topos, and Dolbeault modules. The chapter begins with a short aside on small generalized representations in the affine case, which will be used as intermediary for the study of Dolbeault representations. It then introduces the notion of generalized Dolbeault representation for a small affine scheme and the companion notion of solvable Higgs module, and constructs a natural equivalence between these two categories. It establishes links between these notions and Faltings smallness conditions and relates this to Hyodo's theory. It also describes the Higgs–Tate algebras and concludes with an analysis of the logical links for a Higgs bundle, between smallness and solvability.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

EXCEPTIONAL SEIFERT GROUP ACTIONS ON R

1999 ◽  
Vol 08 (02) ◽  
pp. 241-247 ◽  
Author(s):  
R. ROBERTS ◽  
M. STEIN
Keyword(s):  
Fundamental Group ◽  
Group Action ◽  
Group Actions

We show that every group action on ℝ of the fundamental group of an exceptional Seifert fibered space M corresponds to an action on the order tree determined by the space of leaves of a foliation on M.

Triple Cohomology and the Galois Cohomology of Profinite Groups

1974 ◽  
Vol 1 (6) ◽  
pp. 459-473 ◽  
Author(s):  
D. Gildenhuys ◽  
E. Mackay
Keyword(s):  
Galois Cohomology ◽  
Profinite Groups

scholarly journals Finite group actions on 4-manifolds

1999 ◽  
Vol 66 (3) ◽  
pp. 287-296 ◽  
Author(s):  
Yong Seung Cho
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Group Actions ◽  
Witten Invariant ◽  
Finite Group Actions ◽  
Finite Fundamental Group ◽  
A Finite Group

AbstractLet X be a closed, oriented, smooth 4-manifold with a finite fundamental group and with a non-vanishing Seiberg-Witten invariant. Let G be a finite group. If G acts smoothly and freely on X, then the quotient X/G cannot be decomposed as X1#X2 with (Xi) > 0, i = 1, 2. In addition let X be symplectic and c1(X)2 > 0 and b+2(X) > 3. If σ is a free anti-symplectic involution on X then the Seiberg-Witten invariants on X/σ vanish for all spinc structures on X/σ, and if η is a free symplectic involution on X then the quotients X/σ and X/η are not diffeomorphic to each other.

scholarly journals A non-Archimedean analogue of Teichmüller space and its tropicalization

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Martin Ulirsch
Keyword(s):  
Fundamental Group ◽  
Outer Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Projective Curve ◽  
Deformation Retract ◽  
Logarithmic Geometry ◽  
Strong Deformation ◽  
Classical Construction

AbstractIn this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space$$\overline{{{\mathcal {T}}}}_g$$ T ¯ g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in $$\overline{{\mathcal {T}}}_g$$ T ¯ g .

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