Related papers: Representations of surface groups with finite mapp…

In this note, we prove that for every ordinary genus-2 curve $X$ over a finite field $\kappa$ of characteristic 2 with $\text{Aut}(X/\kappa)=\db{Z}/2\db{Z} \times S_3$, there exist $\text{SL}(2,\kappa\sembrack{s})$-representations of…

We study dismantling properties of the arc, disc and sphere graphs. We prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out(F_n) fixes a filling (resp. simple) clique in the…

Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $\rho:\pi_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $\rho_0:\pi_1(S) \rightarrow…

Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between…

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every…

Let $K$ be an algebraically closed field of characteristic $0$ and let $G$ be a finite cyclic group of order $n$. In this note we prove, using induction on the number of prime divisors of $n$, that $R_K(G)/I \cong \mathbb{Z}[X]/\langle…

We show that for every representation $ \rho : \pi_{1} (S_{g}) \to \text{Isom}(X) $ of the fundamental group of a genus $ g \ge 2 $ surface to the isometry group of a complete $ \text{CAT}(-1) $ metric space $ X $ there exists a Fuchsian…

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is…

Let $\rho$ be a representation of the fundamental group of a punctured surface into $\mathrm{PSL}_2 (\mathbb{C})$ that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates $\rho$ in the simple length…

We show that for a smooth, closed 2-connected manifold $M$ of dimension $d \geq 6$, the topological mapping class group $\pi_0 \mathrm{Homeo}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group $\pi_0…

When $G$ is a connected compact Lie group, and $\pi$ is a closed surface group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset which is a smooth symplectic manifold. This symplectic structure is $Out(\pi)$-invariant…

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of…

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…

We introduce a new approach to representation theory of finite groups that uses some basic algebraic geometry and allows to do all the theory without using characters. With this approach, to any finite group $G$ we associate a finite number…

We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective…

Every representation of the Cuntz algebra $\mathcal{O}_n$ leads to a unitary representation of the Higman-Thompson group $V_n$. We consider the family $\{\pi_x\}_{x\in [0,1[}$ of permutative representations of $\mathcal{O}_n$ that arise…

In this paper, we study the action of finite subgroups of the mapping class group of a surface on the curve complex. We prove that if the diameter of the almost fixed point set of a finite subgroup H is big enough, then the centralizer of H…

Let $F/k$ be a cyclic extension of number fields of prime degree. Let $\rho$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $\pi$ a cuspidal automorphic representation of…

We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of…

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.