Related papers: A zoo of growth functions of mapping class sets

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y \to Y_{(0)}$ which is well-understood. Early on it was found that the induced map $[X,Y] \to [X,Y_{(0)}]$ on sets of mapping classes is…

We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties. Our main result shows that (for sufficiently "nice" random walk measures) a connected, compactly…

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main…

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite-dimensional algebras with polynomial growth, under mild assumptions of the number of relations of given degrees. In case these algebras are finitely…

An important problem in combinatorial noncommutative algebra is to characterize the growth functions of finitely generated algebras (equivalently, semigroups, or hereditary languages). The growth function of every finitely generated,…

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras.…

We conjecture the true rate of growth of the maximum size of the Riemann zeta function and other $L$-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of $L$-functions, and also by…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

For $k,l\ge2$ we consider ideals of edge $l$-colored complete $k$-uniform hypergraphs $(n,\chi)$ with vertex sets $[n]=\{1, 2, \dots n\}$ for $n\in\mathbb{N}$. An ideal is a set of such colored hypergraphs that is closed to the relation of…

We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent…

We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative…

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$.…

We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic.

We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact K\"ahler manifold with nonnegative bisectional…

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for…

In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse…

We construct an increasing, submultiplicative, arbitrarily rapid function which is not equivalent to the growth function of any finitely generated algebra, demonstrating the difficulty in characterizing growth functions in an asymptotic…

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$.…

We characterize the Archimedean lattice-ordered algebras with identity that admit a polynomial growth continuous function calculus. More precisely, for an $n$-tuple $\mathbf{x}=(x_1,\dots,x_n)$ in an Archimedean lattice-ordered algebra $X$…

Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points…