Related papers: Balls in groups: volume, structure and growth

A finitely generated group $\Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the image $\varphi(\Gamma)$ of finite index in $\Gamma$ and the subgroup $\displaystyle…

Gromov claimed, with a sketch of proof, that simply connected nilpotent Lie groups have polynomially bounded filling invariants. The literature establishes this, often with a stronger conclusion where the exponent of polynomiality is…

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the…

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 give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by…

We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ of density at least $(\log\log{N})^{-c}$ contains a nontrivial progression of the form $x,x+y,x+y^2$. This is the first quantitatively effective version of the…

Let $G =<S>$ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$…

We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can…

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined…

Let S be a generating set of a group G. We say that G has FINITE WIDTH relative to S if G=(S\cup S^{-1})^k for a suitable natural number k. We say that a group G is a group of FINITE C-WIDTH if G has finite width with respect to all…

This note constructs a finitely generated group $W$ whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 -- in other words, of non-uniformly exponential growth. This answers a…

The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this paper, we give a particular…

On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the…

We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely,…

Grohe and Marx proved that if G does not contain H as a topological minor, then there exist constants g=O(|V(H)|^4), D and t depending only on H such that G is a clique sum of graphs that either contain at most t vertices of degree greater…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

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…

This study delves into the exploration of the limiting shape theorem for subadditive processes on finitely generated groups with polynomial growth, commonly referred to as virtually nilpotent groups. Investigating the algebraic structures…

Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log n}$, the degree…

Every automaton group naturally acts on the space $X^\omega$ of infinite sequences over some alphabet $X$. For every $w\in X^\omega$ we consider the Schreier graph $\Gamma_w$ of the action of the group on the orbit of $w$. We prove that for…