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Alexey Talambutsa

Publications and source records attributed to Alexey Talambutsa.

9 recordsLinked to original sources
arXiv2025

On the word problem for just infinite groups

We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of relations. Our proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups and proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem and McKinsey--Maltsev theorem. For countably generated presentations of just infinite groups with a recursively enumerable set of relations we show that the word problem is decidable in all cases except locally finite groups without a computable lower bound on subgroup sizes. Finally, we construct presentations of countably generated locally finite groups with recursively enumerable set of relations, for which the word problem is undecidable. Yet, there exist other presentations of these groups, for which the word problem is decidable.

math.GR↗
arXiv2024

A simple algorithm for checking equivalence of counting functions on free monoids

In this note we propose a new algorithm for checking whether two counting functions on a free monoid $M_r$ of rank $r$ are equivalent modulo a bounded function. The previously known algorithm has time complexity $O(n)$ for all ranks $r>2$, but for $r=2$ it was estimated only to be $O(n^2)$. We apply a new approach based on the explicit basis expansion and summation of weighted rectangles, which allows us to construct a much simpler algorithm with time complexity $O(n)$ for any $r\geq 2$. We work in the multi-tape Turing machine model with non-constant-time arithmetic operations.

cs.DM↗
arXiv2021

Efficient computations with counting functions on free groups and free monoids

We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on non-abelian free groups are cohomologous. We work in the multi-tape Turing machine model with non-constant time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear space and time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, i.e. it is as fast as addition of rational numbers (implemented using the Harvey--van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions.

math.GR↗
arXiv2021

On free semigroups of affine maps on the real line

In this note we generalise some of the work of Klarner on free semigroups of affine maps acting on the real line by using a classical approach from geometric group theory (the Ping-Pong lemma). We also investigate the boundaries within which Klarner's necessary condition for a semigroup to be related is applicable.

math.CO↗
arXiv2021

Growth rates of Coxeter groups and Perron numbers

We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.

math.GR↗
arXiv2015

Relations between counting functions on free groups and free monoids

We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n \geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.

math.GR↗
arXiv2015

Minimal exponential growth rates of metabelian Baumslag-Solitar groups and lamplighter groups

We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.

math.GR↗
arXiv2012

Exponential growth rates of free and amalgamated products

We prove that there is a gap between $\sqrt{2}$ and $(1+\sqrt{5})/2$ for the exponential growth rate of free products $G=A*B$ not isomorphic to the infinite dihedral group. For amalgamated products $G=A*_C B$ with $([A:C]-1)([B:C]-1)\geq2$, we show that lower exponential growth rate than $\sqrt{2}$ can be achieved by proving that the exponential growth rate of the amalgamated product $\mathrm{PGL}(2,\mathbb{Z})\cong (C_2\times C_2) *_{C_2} D_6$ is equal to the unique positive root of the polynomial $z^3-z-1$. This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011) 208--217].

math.GR↗