Related papers: On the growth of iterated monodromy groups

Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. We show that the monodromy group of a good…

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence…

We prove a variant of the multidimensional polynomial Szemer\'edi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain…

Quadratic conjecture is a strengthening of oliver's $p$-group conjecture. Let $G$ be a $p$-group of maximal class of order $p^n$. We prove that if $n\le 8$ or $n\ge \max\{2p-6,p+2\}$ then $G$ satisfies Quadratic Conjecture. Hence quadratic…

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…

Given a group G acting on a geodesic metric space, we consider a preferred collection of paths of the space -- a path system -- and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the…

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an…

For the universal family of cyclic covers of projective spaces branched along hyperplane arrangements in general position, we consider its monodromy group acting on an eigen linear subspace of the middle cohomology of the fiber. We prove…

We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain…

We prove that for any circulant matrix $C$ of size $n\times n$ with the monic characteristic polynomial $p(z)$, the spectrum of its $(n-1)\times(n-1)$ submatrix $C_{n-1}$ constructed with first $n-1$ rows and columns of $C$ consists of all…

The conjecture of Valent about the type of Jacobi matrices with polynomially growing weights is proved.

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most…

We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1}) are given by noncommutative Laurent polynomials.

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic…

We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.

The theory of elliptic pairs, as investigated in a paper by Castravet, Laface, Tevelev, and Ugaglia, provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions…

In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…

We introduce the notion of a categorical valuative invariant of polyhedra or matroids, in which alternating sums of numerical invariants are replaced by split exact sequences in an additive category. We provide categorical lifts of a number…

Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of $\mathbb{C}^n$.