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Azer Akhmedov

Publications and source records attributed to Azer Akhmedov.

At least 19 recordsLinked to original sources
arXiv2026

On Dense Subgroups of Homeo(I)

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation in it. In the last section, we demonstrate that finitely generated dense subgroups have infinite girth.

math.GR↗
arXiv2025

A note on the problem of straight-line interpolation by ridge functions

In this paper we discuss the problem of interpolation on straight lines by linear combinations of ridge functions with fixed directions. By using some geometry and/or systems of linear equations, we constructively prove that it is impossible to interpolate arbitrary data on any three or more straight lines by sums of ridge functions with two fixed directions. The general case with more straight lines and more directions is reduced to the problem of existence of certain sets in the union of these lines.

math.CA↗
arXiv2025

On groups of diffeomorphisms of the interval with finitely many fixed points

In [13], it is proved that any subgroup of $\mathrm{Diff}_{+}^{ω}(I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether or not the Girth Alternative holds for subgroups of $\mathrm{Diff}_{+}^{ω}(I)$. In this paper, we answer this question affirmatively for even a larger class of groups of orientation preserving diffeomorphisms of the interval where every non-identity element has finitely many fixed points. We show that every such (irreducible) group is either affine (in particular, metaabelian) or has infinite girth. The proof is based on our study of discrete subgroups of the diffeomorphism group $\mathrm{Diff}_{+}(I)$ which we initiated in [9] and later developed in [1] and [2]; more specifically, our results are obtained by sharpening the tools from the earlier works [1] and [2]. One of the major tools (local transitivity) is heavily exploited in [2] to get an extension of Holders theorem which is crucially used in this paper. We show that local transitivity can be proved for any (up to a conjugacy) non-affine group of irreducible diffeomorphisms with every non-identity element having finitely many fixed points.

math.GR↗
arXiv2024

Big Tiles in Hyperbolic Groups

We prove that if $Γ$ is a word hyperbolic group and $K$ is a finite subset of $Γ$, then $Γ$ admits a tile containing $K$.

math.GR↗
arXiv2023

Girth Alternative for HNN Extensions

We prove the Girth Alternative for a sub-class of the HNN extensions of finitely generated groups. We also produce counterexamples to show that beyond our class, the alternative fails in general.

math.GR↗
arXiv2023

On groups of homeomorphisms of the interval with finitely many fixed points

We strengthen the results of \cite{A1}, consequently, we improve the claims of \cite{A2} obtaining the best possible results. Namely, we prove that if a subgroup $Γ$ of $\mathrm{Diff}_{+}(I)$ contains a free semigroup on two generators then $Γ$ is not $C_0$-discrete. Using this, we extend the Hölder's Theorem in $\mathrm{Diff}_{+}(I)$ classifying all subgroups where every non-identity element has at most $N$ fixed points. In addition, we obtain a non-discreteness result in a subclass of homeomorghisms which allows to extend the classification result to all subgroups of $\mathrm{Homeo}_{+}(I)$ where every non-identity element has at most $N$ fixed points.

math.GR↗
arXiv2017

Non-bi-orderability of $6_2$ and $7_6$

We prove that the knot groups of $6_2$ and $7_6$ are not bi-orderable. These are the only two knot groups up to 7 crossings whose bi-orderability was not known. Our method applies to a very broad class of knots.

math.GR↗
arXiv2015

Finiteness of the topological rank of diffeomorphism groups

For a compact smooth manifold $M$ (with boundary) we prove that the topological rank of the diffeomorphism group Diff$_0^k(M)$ is finite for all $k\geq 1$. This extends a result from [2] where the same claim is proved in the special case of dim M = k = 1.

math.GR↗
arXiv2015

Existence and genericity of finite topological generating sets for homeomorphism groups

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite topological generating sets in related groups. We show that the generic pair of elements in the homeomorphism group $Homeo_+(I)$ generate a dense subgroup of $Homeo_+(I)$. By contrast, if $M$ is any compact connected manifold with boundary other than the interval, we observe that an open dense set of pairs from the associated boundary-fixing homeomorphism group $Homeo(M,\partial M)$ will generate a discrete subgroup. We make similar observations for homeomorphism groups of manifolds without boundary including $\mathbb{S}^1$.

math.GR↗
arXiv2014

Arithmetic Sets in Groups

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.

math.GR↗
arXiv2014

On the height of subgroups of Homeo(I)

In [Bl1], it is proved that a subgroup of $PL_{+}(I)$ has a finite height if and only if it is solvable. We prove the "only if" part for any subgroup of Homeo$_{+}(I)$, and present a construction which indicates a plethora of examples of solvable groups with infinite height.

math.GR↗