Azer Akhmedov

On the girth of finitely generated groups

Abstract We study the notion of girth for finitely generated groups. It is proved that if the girth is ‘small’ then the group necessarily satisfies a law. A general construction is presented which provides examples of groups with infinite girth not containing nonabelian free group. We also prove that SL(n, Z ) has infinite girth.

Chordal and timbral morphologies using Hamiltonian cycles

In this paper, we investigate several musical morphologies that can be represented as paths in abstract graphs. Our examples come from questions posed by composers in the compositional process. In particular, we focus on Hamiltonian paths and cycles, which are central to graph theory. Our results show the circumstances in which such a path exists in the graphs derived from these musical ideas.

Extension of Hölder’s theorem in

We prove that if Γ is subgroup of Diff 1+ + (I) and N is a natural number such that every non-identity element of Γ has at most N fixed points then Γ is solvable. If in addition Γ is a subgroup of Diff +(I) then we can claim that Γ is metaabelian. It is a classical result (essentially due to Hölder, cf.[N1]) that if Γ is a subgroup of Homeo+(R) such that every nontrivial element acts freely then Γ is Abelian. A natural question to ask is what if every nontrivial element has at most N fixed points where N is a fixed natural number. In the case of N = 1, we do have a complete answer to this question: it has been proved by Solodov (not published), Barbot [B], and Kovacevic [K] that in this case the group is metaabelian, in fact, it is isomorphic to a subgroup of the affine group Aff(R). (see [FF] for the history of this result, where yet another nice proof is presented). In this paper, we answer this question for an arbitrary N assuming some regularity on the action of the group. Our main result is the following theorem. Theorem 0.1.(Main Theorem) Let ∈ (0, 1) and Γ be a subgroup of Diff 1+ + (I) such that every nontrivial element of Γ has at most N fixed points. Then Γ is solvable. Assuming a higher regularity on the action we obtain a stronger result Theorem 0.2. Let Γ be a subgroup of Diff +(I) such that every nontrivial element of Γ has at most N fixed points. Then Γ is metaabelian. An important tool in obtaining these results is provided by Theorems B-C from [A]. Theorem B (Theorem C) states that a non-solvable (non-metaabelian) subgroup of Diff 1+ + (I) (of Diff 2 +(I)) is non-discrete in C0 metric. Existence of C0-small elements in a group provides effective tools in tackling the problem. Such tools are absent for less regular actions, and the problem for Homeo+(I) (even for Diff+(I)) still remains open. Basic Notations: Throughout this paper, G will denote the group Diff 1+ + (I) where ∈ (0, 1). We let Γ ≤ G. For every g ∈ Γ, Fix(g) will denote the set of fixed of points of g in (0, 1). A fixed point 1 ar X iv :1 30 8. 02 50 v2 [ m at h. G R ] 2 J un 2 01 4

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].

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.

On the height of subgroups of Homeo+(I)

Abstract In [J. Lond. Math. Soc. (2) 78 (2008), no. 2, 352–372], it is proved that a subgroup of PL+(I) has 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.

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.