The topology of moduli spaces of free group representations

Florentino, Carlos; Lawton, Sean

doi:10.1007/s00208-009-0362-4

Cited by 41 publications

(73 citation statements)

References 33 publications

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“…In particular, if K ≤ G is a maximal compact subgroup, then DK is also compact and applying this result with H = DK , we find that the conjugation quotient DK /DK is simply connected. Now, the main result of [10] gives a homotopy equivalence between the conjugation quotients DG/ /DG and DK /DK , so we conclude that DG/ /DG is simply connected as well. It follows that all elements in π 1 (DG) all map to 0 in π 1 (X 0 (DG)) by commutativity.…”

Section: Lemma 23 Let Be Exponent-canceling and G A Connected Reductmentioning

confidence: 95%

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Fundamental groups of character varieties: surfaces and tori

2015

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Section: Lemma 23 Let Be Exponent-canceling and G A Connected Reductmentioning

confidence: 95%

“…On the other hand, as shown in [5,10,11], when G is a real reductive Lie group and is free (Abelian or non-Abelian), X (G) is homotopy equivalent to X (K ).…”

Section: Conjecture 27mentioning

confidence: 99%

Fundamental groups of character varieties: surfaces and tori

2015

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“…Indeed, it is shown in Florentino-Lawton-Ramras [10,Proposition 3.4] that Hom(π 1 Σ, GL(n))/GL(n) (which is usually non-Hausdorff) deformation retracts to the geometric invariant theory quotient Hom(π 1 Σ, GL(n))/ /GL(n), which in turn deformation retracts to the subspace of unitary characters by Florentino-Lawton [8,9] (see also Bergeron [2]). Hence in these cases, Proposition 7.6 applies to spherical families of general linear representations as well.…”

Section: σKumentioning

confidence: 99%

The homotopy groups of a homotopy group completion

Ramras

2019

Isr. J. Math.

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Let M be a topological monoid with homotopy group completion ΩBM . Under a strong homotopy commutativity hypothesis on M , we show that π k (ΩBM ) is the quotient of the monoid of free homotopy classes [S k , M ] by its submonoid of nullhomotopic maps.We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.

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“…(See Li [124] and Rapinchuk-Benyash-Krivetz-Chernousov [141].) Recently Florentino and Lawton [58] have determined the homotopy type of Hom(Γ, G)//G when Γ is free and G is a complex reductive group.…”

Section: Theorem 1 Equality Holds In (3) If and Only If ρ Is A Discrmentioning

confidence: 99%

“…(See Li [124] and .) Recently Florentino and Lawton [58] have determined the homotopy type of Hom(Γ, G)//G when Γ is free and G is a complex reductive group.This simple picture becomes much more intricate and fascinating for higher dimensional noncompact real Lie groups; the most effective technique so far has been the interpretation in terms of Higgs bundles and the use of infinite-dimensional Morse theory; see for a survey of some recent results on the components when G is a simple real Lie group. …”

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Locally Homogeneous Geometric Manifolds

Goldman

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract.Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.Mathematics Subject Classification (2000). Primary 57M50; Secondary 57N16.Keywords. connection, curvature, fiber bundle, homogeneous space, Thurston geometrization of 3-manifolds, uniformization, crystallographic group, discrete group, proper action, Lie group, fundamental group, holonomy, completeness, development, geodesic, symplectic structure, Teichmüller space, Fricke space, hypebolic structure, Riemannian metric, Riemann surface, affine structure, projective structure, conformal structure, spherical CR structure, complex hyperbolic structure, deformation space, mapping class group, ergodic action. Historical backgroundWhile geometry involves quantitative measurements and rigid metric relations, topology deals with the loose quantitative organization of points. Felix Klein proposed in his 1872 Erlangen Program that the classical geometries be considered as the properties of a space invariant under a transitive Lie group action. Therefore one may ask which topologies support a system of local coordinates modeled on a fixed homogeneous space X = G/H such that on overlapping coordinate patches, the coordinate changes are locally restrictions of transformations from G. W. GoldmanIn this generality this question was first asked by Charles Ehresmann [55] at the conference "Quelques questions de Geométrie et de Topologie," in Geneva in 1935. Forty years later, the subject of such locally homogeneous geometric structures experienced a resurgence when W. Thurston placed his 3-dimensional geometrization program [158] in the context of locally homogeneous (Riemannian) structures. The rich diversity of geometries on homogeneous spaces brings in a wide range of techniques, and the field has thrived through their interaction.Before Ehresmann, the subject may be traced to several independent threads in the 19th century:• The theory of monodromy of Schwarzian differential equ...

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The topology of moduli spaces of free group representations

Cited by 41 publications

References 33 publications

Fundamental groups of character varieties: surfaces and tori

Fundamental groups of character varieties: surfaces and tori

The homotopy groups of a homotopy group completion

Locally Homogeneous Geometric Manifolds

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