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

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28 papers · 1 filter

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.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.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.

math.GR2013★ 2 cited

Extension of Hölder's Theorem in Diff_{+}^{1+ε}(I)

Azer Akhmedov

We prove that if Γis subgroup of Diff_{+}^{1+ε}(I) and N is a natural number such that every non-identity element of Γhas at most N fixed points then Γis solvable. If in addition Γ…

math.GR2012

Amenable subgroups of Homeo(R) with large characterizing quotients

Azer Akhmedov

We construct a finitely generated solvable subgroup of Homeo(R) with a non-metaabelian characterizing quotient.

math.GR2012★ 1 cited

A weak Zassenhaus lemma for discrete subgroups of Diff(I)

Azer Akhmedov

We prove a weaker version of Zassenhaus Lemma (also known as Margulis Lemma) for subgroups of Diff(I). We also show that a group with commutator subgroup containing a free subsemig…

math.GR2011★ 4 cited

Girth Alternative for Subgroups of PL_o(I)

Azer Akhmedov

We prove the Girth Alternative for finitely generated subgroups of PL_o(I). We also prove that a finitely generated subgroup of Homeo(I) which is sufficiently rich with hyperbolic-…

Margulis-Soifer theorem for one-relator groups

A note on maximal subgroups of countable groups

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

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

Extension of Hölder's Theorem in Diff_{+}^{1+ε}(I)

Amenable subgroups of Homeo(R) with large characterizing quotients

A weak Zassenhaus lemma for discrete subgroups of Diff(I)

Girth Alternative for Subgroups of PL_o(I)

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