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We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…
The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper we obtain 2-step growth results for representations of such groups $G$ (including…
We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main…
Consider an arithmetic group $\mathbf{G}(O_S)$, where $\mathbf{G}$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of $S$-integers $O_S$ of a number field $K$ with…
We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties…
We study Sushchansky p-groups. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on p-ary tree are not level-transitive and we describe their orbit trees.…
We study the growth of typical groups from the family of $p$-groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound.…
Schreier formula for the rank of a subgroup of finite index of a finitely generated free group $F$ is generalized to an arbitrary (even infinitely generated) subgroup $H$ through the Schreier transversals of $H$ in $F$. The rank formula may…
We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to…
We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). The class of these groups contains extraordinary examples like Grigorchuk group, which is known to be…
We give sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's theorem on groups with polynomial growth. Precisely, we show that for every non-negative integer d there exists $c=c(d)>0$ such that if $G$ is a group with…
The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric…
We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For…
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…
This paper gives asymptotic formulas for the subgroup growth and maximal subgroup growth of all Baumslag-Solitar groups.
Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_{R} denote the canonical right invariant Riemannian metric on G. For v in R^{n} let G_{v} = {g in G : g(v) = v}. We obtain algebraically defined upper…
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric…
We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible…
It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces defined by Witten and Kontsevich are polynomials in the adjusted Miller-Morita-Mumford classes. The leading coefficient was…
The asymptotic shape of randomly growing radial clusters is studied. We pose the problem in terms of the dynamics of stochastic partial differential equations. We concentrate on the properties of the realizations of the stochastic growth…