arXiv subjects

Azer Akhmedov

Publications and source records attributed to Azer Akhmedov.

33 records · Page 2Linked to original sources

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↗

arXiv2015

Cayley graphs with an infinite Heesch number

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

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↗

arXiv2014

Girth Alternative for Subgroups of PL_o(I)

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-like elements has infinite girth.

math.GR↗

arXiv2014

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

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 Γis a subgroup of Diff_{+}^{2}(I) then we can claim that Γis metaabelian.

math.GR↗

arXiv2014

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

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 $\mathrm{Diff}_{+}^{ω}(\mathbb{S}^1)$. We also prove that a branch group does not embed into $\mathrm{Diff}_{+}^{ω}(\mathbb{S}^1)$.

math.GR↗

arXiv2014

Balance in Random Trees

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.CO↗

arXiv2013

A new metric criterion for non-amenability III: Non-amenability of R.Thompson's group F

We present new metric criteria for non-amenability and discuss applications. The main application of the results of this paper is the proof of non-amenability of R.Thompson's group F. This is a continuation of the series of papers on our criteria for non-amenability. The current paper is independent of other papers in the series.

math.GR↗

arXiv2013

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

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.GR↗

arXiv2013

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

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 subsemigroup does not admit a C_0-discrete faithful representation in Diff(I).

math.GR↗

arXiv2013

Good Modulating Sequences for the Ergodic Hilbert Transform

This article investigates classes of bounded sequences of complex numbers that are universally good for the ergodic Hilbert transform in L_p-spaces, 2\leq p\leq \infty : The class of bounded Besicovitch sequences satisfying a rate condition is among such sequence classes.

math.DS↗

arXiv2010

Free limits of Thompson's group $F$

We produce a sequence of markings $S_k$ of Thompson's group $F$ within the space ${\mathcal G}_n$ of all marked $n$-generator groups so that the sequence $(F,S_k)$ converges to the free group on $n$ generators, for $n \geq 3$. In addition, we give presentations for the limits of some other natural (convergent) sequences of markings to consider on $F$ within ${\mathcal G}_3$, including $(F,\{x_0,x_1,x_n\})$ and $(F,\{x_0,x_1,x_0^n\})$.

math.GR↗

arXiv2010

On free discrete subgroups of Diff(I)

We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.

math.GR↗

arXiv2009

A new metric criterion for non-amenability I

By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent.

math.GR↗