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

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…

Let F be the (Thompson's) group < x_0, x_1 | [x_0x_1^-1, x_0^-ix_1 x_0^i], i=1,2 >. We study the structure of F-limit groups. Let G_n= < y_1,..., y_m, x_0,x_1 | [x_0x_1^-1,x_0^-1x_1x_0],[x_0x_1^-1,x_0^-2x_1x_0^2], y_j^-1g_j,n(x_0,x_1),…

In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.

We show that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a…

We give a criterion for an HNN extension of a finite $p$-group to be residually $p$.

We introduce a new class of Abelian groups which lies strictly between the classes of co-Hopfian groups and Dedekind-finite groups, calling these groups {\it Bassian-finite}. We prove the surprising fact that in the torsion case the…

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from…

We prove that for any non-symmetric irreducible divisible convex set, the proximal limit set is the full projective boundary.

An upper bound of composition series of groups of finite order is obtained. The bound is a nontrivial bound and so far best possible.

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik-Chervonenkis density. Furthermore, strong abelian groups are…

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…

We develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus-1 curves. We use this approach to prove a purely elementary criterion for proving cases of…

In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.

We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups.

We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the…

We give the first example of a non-linear residually finite 1-related group: < a, t | a^{t^2}=a^2>.

In this paper, we introduce the relative $n$-tensor nilpotent degree of a finite group $G$ with respect to a subgroup $H$ of $G$. The aim of this paper is to investigate this concept and give some results on this topic.

We show that the algebraic local fundamental group of any klt singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

In this paper we prove that a free nilpotent group of finite rank is transitive self-similar. In contrast, we prove that a free metabelian group of rank $r \geq 2$ is not transitive self-similar.

We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be…