Representations Of Fundamental Groups Research Articles

Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let Sigma _{g} denote a topological surface of genus gge 2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of pi _{1}(Sigma _{g}) under a random representation of pi _{1}(Sigma _{g}) into mathsf {SU}(n). Each such expected value involves a contribution from all irreducible representations of mathsf {SU}(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

A note on Higgs-de Rham flows of level zero

The notion of Higgs-de Rham flows was introduced by Lan et al. (2019), as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In this paper we investigate a small part of this theory, and study those Higgs-de Rham flows which are of level zero. We improve the original definition of level-zero Higgs-de Rham flows (which works for general levels), and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic, which generalizes a classical results of Katz (1973). We compare the deformation theories of two sides in the correspondence, and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.

Fundamental classes of 3-manifold groups representations in SL(4,R)

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of [Formula: see text]-manifolds to [Formula: see text] that factor over [Formula: see text], in particular for those factoring over the isomorphism [Formula: see text]. We also discuss consequences for the number of connected components of [Formula: see text]-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern–Simons invariant in their [Formula: see text]-character varieties.

Cohomology jump loci of differential graded Lie algebras

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

Twisted linking numbers via representations of fundamental groups

Using representations of fundamental groups, we introduce the concept of twisted linking numbers. If the corresponding representation is trivial, the twisted linking number coincides with the linking number. In the case of a nontrivial representation, however, this is not necessarily true, which implies that the twisted linking number can detect the nontriviality of an embedding of S 1 . An example is given for which the linking number is trivial but the twisted linking number is not.

Representations of fundamental groups whose Higgs bundles are pullbacks

Representations of fundamental groups whose Higgs bundles are pullbacks

On ideal points of deformation curves of hyperbolic 3-manifolds with one cusp

BY a hyperbolic 3-manifold we will mean an oriented complete hyperbolic 3-manifold of finite volume. In [I], Culler and Shalen investigated varieties of characters of representations of fundamental groups of hyperbolic 3-manifolds into S&(C). They showed that there corresponds an incompressible and boundary incompressible surface in the hyperbolic 3-manifold to each ideal point of an algebraic curve in its character variety. They used valuation theory and Bass-Serre’s theory of group actions on trees. The purpose of this paper is to give an explicit understanding of the restriction of the above gcncral theory to the most interesting special case of hyperbolic knot complements. Instead of character varieties, we USC deformation curves dcfincd by Thurston’s idcal triangulations and WC adopt the combinatorics developed by Nrumann and Zagicr in [3]. Our construction of incompressible surfaces is simpler and more explicit. Let N bc a hyperbolic 3-manifold with one cusp. Topologically N is homeomorphic to the interior of a compact 3-manifold with boundary homeomorphic to the 2-torus. We assume that N is decomposed into a finite union of ideal tetrahedra with mutually disjoint interiors. Such a decomposition is called an ideal triangulation of N. Associated to an idcal triangulation of N. there is an alTine algebraic curve C called a deformation curve ($2). In a neighborhood of the point corresponding to the complete hyperbolic structure, the points of C represent (incomplete) smooth hyperbolic structures on N. In 93, we dcfinc ideal points of C. We see that there corresponds a slope (an element of H,(?N) mod scalar multiplication) to each ideal point of C. We call it the boundary slope of the ideal point. In $4, for each idcal points of C, we construct a properly embedded surface in N whose boundary slope is that of the ideal point by patching together some copies of Thurston’s twisted square ([2]) contained in the ideal tetrahedra. In $5, we show that the splitting of N by the above surface is non-trivial and we can obtain an incompressible and boundary-incompressible surface in N after performing compressions on it.

Congruence Subgroups of the Picard Group

The Picard group Γ = PSL2 (Z [i]) is the group of linear fractional transformationswith ad – bc = ± 1 and a, b, c, d Gaussian integers.Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.