Related papers: Growth of permutational extensions

We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In…

In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let $\Sigma_{g}$ denote a…

We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$-torsion element. We define…

We first show that a class of pro-$p$ branch groups including the Grigorchuk group and the Gupta-Sidki groups all have subgroup growth type $n^{\log n}$. We then introduce the notion of orbit growth and use it to construct extensions of the…

We provide a class of non-contracting groups containing an infinite family of fractal and weakly regular branch groups, and study certain properties including abelianization, just infiniteness, and word problem. We present an example of a…

We calculate full asymptotic expansions of prime-independent multiplicative functions on additive arithmetic semigroups that satisfy a strong form of Knopfmacher's axioms. When applied to the semigroup of unlabeled graphs, our method yields…

Methods from additive number theory are applied to construct families of finitely generated linear semigroups with intermediate growth.

This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of…

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose…

What is the true order of growth of torsion in the cohomology of an arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field $F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle \sigma \rangle.$…

We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian…

Let $n$ be an integer greater than $1$. We consider a group presented as $G(p_1,p_2,\dots,p_n)=\langle x_1,x_2,\dots, x_n \mid x_1^{p_1} =x_2^{p_2}=\cdots =x_n^{p_n} \rangle$, with integers $p_1,p_2,\dots,p_n$ satisfying $2 \leq p_1 \leq…

In [K. Bou-Rabee, B. Seward, J. Reine Angwe. Math. 2016] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups $G$ whose residual finiteness growth function $\mathcal{F}_G$ can be at least as fast as any…

We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in…

Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved…

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate…

Let $W$ be a finite group and $T$ be an abelian group. Consider an extension $0 \ra T \ra N \ra W \ra 0$. For a smooth projective curve $X$, we give a precise description of the fiber of the quotient by $T$ map $q_T: \cM_X(N) \ra \cM_X(W)$…

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same…