Representation varieties of the fundamental groups of non-orientable surfaces

Benyash-Krivets, Valerii Vatslavovich; Chernousov, Vladimir

doi:10.1070/sm1997v188n07abeh000242

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…These dimensions agree with those given for the characteristic zero case in [42] for oriented groups and in [3] for non-oriented groups. In fact it is well known that the dimension of a variety in characteristic zero coincides with the dimension of its reduction modulo p for all large primes p, and so Theorem 1.8 provides an alternative proof of the characteristic zero dimension results in [42,3].…”

Section: ])supporting

confidence: 78%

“…Note that since the limit points of {ζ G (1) : G finite simple} are 1, 3 2 and 2, it follows that with the above notation, c n (G) < Cn for all finite simple groups G, where C is an absolute constant, and sharper results follow for G = L 2 (q) -see Corollary 2.7. Theorem 1.1 can be extended to the case where G is a nearly simple group -that is, F * (G) is quasisimple (see Theorem 2.8).…”

Section: ])mentioning

confidence: 99%

“…In fact it is well known that the dimension of a variety in characteristic zero coincides with the dimension of its reduction modulo p for all large primes p, and so Theorem 1.8 provides an alternative proof of the characteristic zero dimension results in [42,3].…”

Section: ])mentioning

confidence: 99%

“…This has a natural structure as an algebraic variety defined over the prime subfield of K, and has been extensively studied in the case where K has characteristic zero and Γ is a surface group (see [35,42,3,12]). However, not much seems to be known in positive characteristic.…”

Section: ])mentioning

confidence: 99%

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Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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Let Γ be a Fuchsian group of genus at least 2 (at least 3 if Γ is non-oriented). We study the spaces of homomorphisms from Γ to finite simple groups G, and derive a number of applications concerning random generation and representation varieties.Precise asymptotic estimates for |Hom(Γ, G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G| μ(Γ)+1+o(1) , where μ(Γ) is the measure of Γ. We then prove that a randomly chosen homomorphism from Γ to G is surjective with probability tending to 1 as |G| → ∞. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties Hom(Γ,Ḡ), whereḠ is GL n (K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic.A key ingredient of our approach is character theory, involving the study of the 'zeta function' ζ G (s) = χ(1) −s , where the sum is over all irreducible complex characters χ of G.

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Section: ])supporting

confidence: 78%

Section: ])mentioning

confidence: 99%

Section: ])mentioning

confidence: 99%

Section: ])mentioning

confidence: 99%

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Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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“…) with a complement of high codimension. See [82] for an affirmative result in the case G = SU(n), G C = SL(n, C).…”

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confidence: 95%

Non-orientable surfaces and electric-magnetic duality

2018

J. High Energ. Phys.

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We consider the reduction along two compact directions of a twisted N = 4 gauge theory on a 4-dimensional orientable manifold which is not a global product of two surfaces but contains a non-orientable surface. The low energy theory is a sigma-model on a 2-dimensional worldsheet with a boundary which lives on branes constructed from the Hitchin moduli space of the non-orientable surface. We modify 't Hooft's notion of discrete electric and magnetic fluxes in gauge theory due to the breaking of discrete symmetry and we match these fluxes with the homotopy classes of maps in the sigma-model. We verify the mirror symmetry of branes as predicted by S-duality in gauge theory.

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Conditions of smoothness of moduli spaces of flat connections and of character varieties

Wilkin

2018

Math. Z.

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We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches.2010 Mathematics Subject Classification. Primary 58D27; Secondary 57M05 20F05.

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Representation varieties of the fundamental groups of non-orientable surfaces

Cited by 10 publications

References 11 publications

Fuchsian groups, finite simple groups and representation varieties

Fuchsian groups, finite simple groups and representation varieties

Non-orientable surfaces and electric-magnetic duality

Conditions of smoothness of moduli spaces of flat connections and of character varieties

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