Related papers: The 4-string Braid group $B_4$ has property RD and…

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the…

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group $\mathrm{Mp}_4$ of rank $2$.

Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid…

In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension…

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined…

The mod 4 braid group, $\mathcal{Z}_{n}$, is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements. Kordek and Margalit proved $\mathcal{Z}_{n}$ is an extension of the…

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$…

As the 3-string braid group B(3) and the modular group PSL(2,Z) are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most'…

We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which…

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic…

In this article, we study the properties of profinite geometric iterated monodromy groups associated to polynomials. Such groups can be seen as generic representations of absolute Galois groups of number fields into the automorphism group…

In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are very strongly fractal, not regular branch, and of exponential growth.…

In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for $SL_3$ over a non archimedean local field $F.$ In this paper, we extend his results to $Sp_4$ and therefore to any connected almost $F$-simple algebraic group with…

We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Brou\'e, Malle and Rouquier and has for consequence that this algebra is a flat…

We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for…

Let $B'_n$ be the commutator subgroup of the braid group $B_n$. We prove that $Aut(B'_n)=Aut(B_n)$ for $n\ge4$. This answers a question asked by Vladimir Lin.

Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a Differential…

The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced…

An automorphism $u$ of a vector space is called unipotent of index $2$ whenever $(u-\mathrm{id})^2=0$. Let $b$ be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space $V$ over a field $\mathbb{F}$ of characteristic…

The n-string braid group of a graph X is defined as the fundamental group of the n-point configuration space of the space X. This configuration space is a finite dimensional aspherical space. A. Abrams and R. Ghrist have conjectured that…