Related papers: Uniform Exponential Growth of Polycyclic Groups

A free-by-cyclic group can often be viewed as a mapping torus of a free group automorphism (monodromy) in multiple ways. What dynamical properties must these monodromies share, and to what extent are they invariant under quasi-isometries?…

Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees…

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…

An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is…

We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.

Let $G$ be a torsion-free hyperbolic group and $\alpha$ an automorphism of $G$. We show that there exists a canonical collection of subgroups that are polynomially growing under $\alpha$, and that the mapping torus of $G$ by $\alpha$ is…

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z}…

We prove the exponential growth of product replacement graphs for a large class of groups. Much of our effort is dedicated to the study of product replacement graphs of Grigorchuk groups, where the problem is most difficult.

We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…

One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010),…

We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different…

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 introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the…

We consider the weight w: 1<w<T on the unit circle and prove that the corresponding orthonormal polynomials can grow.

In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that…

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we…

We announce numerous new results in the theory of orthogonal polynomials on the unit circle.

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of…