We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators ${}_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators ${\Delta }_{ij}^k$ as $log\left(\right|k|+1)$ instead of simply $\left|k\right|$. The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism. Our main...

Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $\left|G\right|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma \left(G\right)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma \left(G\right)=\Gamma \left(B_n\left(5\right)\right)$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_n\left(5\right)$ or $C_n\left(5\right)$.

We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ contains all subextensions of degree less than $d$ if and only if $d\le 4$. We prove that for $d\>2$ there is no bound $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of Bombieri and...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\>2$ and equal to the braid group of the sphere $ℂ$$P^{}$

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension ${dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute ${dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.