A. Akhmedov

activity

20092026

most citedGirth Alternative for Subgroups of PL_o(I)

4 citations · 14 across the 27 of their papers we have counts for

collaborators

33 papers

math.GT2026

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Anar Akhmedov, Azer Akhmedov

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p, q \geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has…

math.GR2026

Margulis-Soifer theorem for one-relator groups

Azer Akhmedov, Atabay Mahmudov

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present…

math.GR2026

A note on maximal subgroups of countable groups

Azer Akhmedov

We show that for any countable group, the set of maximal subgroups is either countable or has cardinality $2^{ℵ_{0}}$ .

math.GT2026

Nonabelian Torus Exteriors in $S^{4}$ and Torus Replacement in Four-Manifolds

Anar Akhmedov, Azer Akhmedov

We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kan…

math.CA2024

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

Azer Akhmedov, Vugar Ismailov

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 li…

math.GR2024

Irreducible discrete subgroups in products of simple Lie groups

Azer Akhmedov

We produce an example of an irreducible discrete subgroup in the product $S L (2, R) \times S L (2, R)$ which is not a lattice. This answers a question asked in [15].

math.GR2023

Big Tiles in Hyperbolic Groups

Azer Akhmedov

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.GR2023

Non-bi-orderable one-relator groups without generalized torsion

Azer Akhmedov, James Thorne

We construct examples of non-bi-orderable one-relator groups without generalized torsion. This answers a question asked in [2].

math.GR2023

Examples of left-orderable and non-left-orderable HNN extensions

Azer Akhmedov, Cody Martin

We prove that an HNN extension of a torsion-free nilpotent group is left-orderable. We also construct examples of non-left-orderable HNN extensions of left-orderable groups

math.GR2022

Girth Alternative for HNN Extensions

Azer Akhmedov, Pratyush Mishra

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 fa…

math.GR2016

Groups not acting on compact metric spaces by homeomorphisms

Azer Akhmedov

We show that the direct sum of uncountably many non-Abelian groups does not embed into the group of homeomorphisms of a compact metric space.

math.GR2015

Non-bi-orderability of $6_{2}$ and $7_{6}$

Azer Akhmedov, Cody Martin

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…

math.GR2015

Finiteness of the topological rank of diffeomorphism groups

Azer Akhmedov

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 fr…

math.GR2015

Existence and genericity of finite topological generating sets for homeomorphism groups

Azer Akhmedov, Michael P. Cohen

We show that the topological groups $D i f f_{+}^{1} (I)$ and $D i f f_{+}^{1} (S^{1})$ of orientation-preserving $C^{1}$ -diffeomorphisms of the interval and the circle, respectively,…

math.GR2015

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

Azer Akhmedov

In [13], it is proved that any subgroup of $Diff_{+}^{ω} (I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does n…

math.GR2015

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

Azer Akhmedov

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{Di…

math.GR2014

Cayley graphs with an infinite Heesch number

Azer Akhmedov

We construct a 2-generated group $Γ$ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.

math.GR2014

Arithmetic Sets in Groups

Azer Akhmedov, Damiano Fulghesu

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.GR2014

Some applications of Hölder's theorem in groups of analytic diffeomorphisms of 1-manifolds

Azer Akhmedov, Michael P. Cohen

We obtain a simple obstruction to embedding groups into the analytic diffeomorphism groups of 1-manifolds. Using this, we classify all RAAGs which embed into $\mathrm{Diff}_{+}^{ω}…

math.CO2014

Balance in Random Trees

Azer Akhmedov, Warren Shreve

We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly $k$ -balanced for any $k \geq 3$ .

math.GR2014

On the height of subgroups of Homeo(I)

Azer Akhmedov

In [Bl1], it is proved that a subgroup of $P L_{+} (I)$ 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…

math.GR2013

On Dense Subgroups of Homeo(I)

Azer Akhmedov

We prove that a dense subgroup of $Homeo_{+} (I)$ is not elementary amenable. We also show that the topological group $Homeo_{+} (I)$ does not satisfy the Stability…

math.GR2013

Questions and Remarks on Discrete and Dense Subgroups of Diff(I)

Azer Akhmedov

We are raising questions on discrete and dense subgroups of Diff(I). Most of the questions are around the problems discussed in [A1]-[A4].

math.GR2013

Non-amenability of R.Thompson's group F

Azer Akhmedov

We present a proof of non-amenability of R.Thompson's group F.

Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Margulis-Soifer theorem for one-relator groups

A note on maximal subgroups of countable groups

Nonabelian Torus Exteriors in S4 and Torus Replacement in Four-Manifolds

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

Irreducible discrete subgroups in products of simple Lie groups

Big Tiles in Hyperbolic Groups

Non-bi-orderable one-relator groups without generalized torsion

Examples of left-orderable and non-left-orderable HNN extensions

Girth Alternative for HNN Extensions

Groups not acting on compact metric spaces by homeomorphisms

Non-bi-orderability of 62​ and 76​

Finiteness of the topological rank of diffeomorphism groups

Existence and genericity of finite topological generating sets for homeomorphism groups

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

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

Cayley graphs with an infinite Heesch number

Arithmetic Sets in Groups

Some applications of Hölder's theorem in groups of analytic diffeomorphisms of 1-manifolds

Balance in Random Trees

On the height of subgroups of Homeo(I)

On Dense Subgroups of Homeo(I)

Questions and Remarks on Discrete and Dense Subgroups of Diff(I)

Non-amenability of R.Thompson's group F

Nonabelian Torus Exteriors in $S^{4}$ and Torus Replacement in Four-Manifolds

Non-bi-orderability of $6_{2}$ and $7_{6}$