Related papers: Groups with a polynomial dimension growth

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…

We begin a generalized study of sum-product type phenomenon in different fields by considering pairs $P(x,y)$ and $Q(x,y)$ of two variable polynomials that simultaneously exhibit small symmetric expansion. Our first result is that such…

We prove the following theorem: Let $\pi$ be the fundamental group of a finite graph of groups with finitely generated vertex groups $G_v$ having asdim $G_v\le n$ for all vertices $v$. Then asdim$\pi\le n+1$. This gives the best possible…

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$…

We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon is an absolute positive real number, or AAA=G. As a corollary we get that the…

In this paper, we introduce several notions of "dimension" of a finite group, involving sizes of generating sets and certain configurations of maximal subgroups. We focus on the inequality $m(G) \leq \mathrm{MaxDim}(G)$, giving a family of…

We present a characterization of the finite groups in which all real classes have prime powers size.

Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively.…

We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually…

In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups…

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…

We consider analogues of Grigorchuk-Gupta-Sidki (GGS-)groups acting on trees of growing degree; the so-called growing GGS-groups. These groups are not just infinite and do not possess the congruence subgroup property, but many of them are…

We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G…

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of…

We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually…

We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of…

We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient…

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ generated by a suitable set of tame automorphisms.…

A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with…