Related papers: Note on the Cantor-Bendixson rank of limit groups

We show that the possible Cantor-Bendixson ranks of countable SFTs are exactly the finite ordinals and ordinals of the form $\lambda + 3$, where $\lambda$ is a computable ordinal. This result was claimed by the author in his PhD…

The goal of this paper is to show the following result: For every integer $n\geq 2$ there is a countable orderable group such that its space of orders is countable and has Cantor-Bendixson rank $n$. We show this by explicitly constructing a…

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (G_n) of 2-generated, finitely presented, virtually metabelian groups of…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

A Cantor minimal system is of finite topological rank if it has a Bratteli-Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set…

We prove that the set of limit groups is recursive, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore,…

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite fields and we establish new asymptotical and not asymptotical upper bounds about it.

The `upper rank' of a group is the supremum of the (Pr\"{u}fer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the…

We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we…

This paper contains a stronger version of a final identification theorem for the `generic' groups of finite Morley rank.

A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the…

The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite…

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles…

In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…

We prove that the outer automorphism group of a free group of countably infinite rank is complete.

We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.