The 3rd JNU-KAIST Geometric Topology Fair: Introductory lectures

Anosov groups acting on symmetric spaces by isometries have been intensively studied for the last twenty years.

Anosov groups are a generalized notion of convex cocompact groups acting on symmetric spaces of rank 1 to higher rank. We will review geometric features of symmetric spaces of higher rank, and look at the definition of Anosov groups, their key properties and applications, and then discuss recent results and research problems on Anosov groups.

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A group is flawed if its moduli space of G-representations is homotopic to its moduli space of K-representations for all reductive affine algebraic groups G with maximal compact subgroup K.  In this talk we discuss this definition, and associated examples, theorems, and conjectures. This work is in collaboration with Carlos Florentino. (Based on arXiv:2012.08481)

Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. In particular there are many connections between Coxeter groups and geometry. This lecture particularly focuses on how one can use Coxeter groups to construct interesting examples of reflection groups in convex real projective geometry.

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The main objective of the talk will be to define $G$-character variety of a finitely presented group $\Gamma$ where $G$ is a reductive affine algebraic group. Specifically, I will discuss the example of $\mathrm{SL}_2(\mathbb{C})$ character varieties of $F_2$, the free group of rank $2$ given by Fricke-Vogt Theorem. I will also briefly talk about the algorithm by Lawton to compute the $\mathrm{SL}_2(\mathbb{C})$ character variety of a finitely generated group $\Gamma$.

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If $G$ is a reductive algebraic group over $\mathbb{Z}$, the $G$-character variety of a finitely presented group $\Gamma$ parametrizes the set of closed conjugation orbits in Hom($\Gamma,G$).  The group of outer automorphisms, $Out(\Gamma)$, naturally acts on the character variety.   The dynamics of this action on the finite field points of character variety is particularly interesting. We explore the transitivity properties of this action.  Specifically, we show that when $\Gamma$ is of free type, that is, $Aut(\Gamma)$ has certain desirable properties, the action is transitive on the set of epimorphisms from $\Gamma$ to $G$. We also look at the $\mathrm{SL}_2(\mathbb{F}_q)$-character variety of $\mathbb{Z}^r$, determined by the set of all $r$-tuples of matrices that commute pairwise, that is being acted upon by the group $\mathrm{GL}_r(\mathbb{Z})$. 

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The adjoint Reidemeister torsion is a 3-manifold invariant having fruitful interaction between hyperbolic geometry and quantum field theory. Regarding the adjoint Reidemeister torsion as a rational function on the character variety, it is conjectured that the adjoint Reidemeister torsion of a hyperbolic 3-manifold admits a vanishing identity. In this talk, we briefly recall the definition of adjoint Reidemeister torsion, and discuss the above conjecture in terms of jacobian of the character variety and the residue theorem.

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