Related papers: Asymptotic Dimension of Discrete Groups

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…

We compute the Hausdorff dimension of the closure of the generalized Brunner-Sidki-Vieira group acting on the $m$-adic tree for $m\ge 2$, providing the first examples of self-similar topologically finitely generated closed subgroups of…

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by…

We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and…

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.

We establish some basic theorems in dimension theory and absolute extensor theory in the coarse category of metric spaces. Some of the statements in this category can be translated in general topology language by applying the Higson corona…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

Bessel and modified Bessel functions of imaginary order $i\nu$ ($\nu >0$) are studied. Asymptotic expansions are derived as $\nu \to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with…

We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new…

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the…

We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem

We show how a recent result of Hrushovsky implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent.

We study the subgroup structure of the infinite torsion $p$-groups defined by Gupta and Sidki in 1983. In particular, following results of Grigorchuk and Wilson for the first Grigorchuk group, we show that all infinite finitely generated…

We prove that for any discrete group $G$ with finite $\mathfrak{F}$-cohomological dimension, the Gorenstein cohomological dimension equals the $\mathfrak{F}$-cohomological dimension. This is achieved by constructing a long exact sequence of…

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are…

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$…

In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups…

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion…

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…

We show that the discrete Heisenberg group has unbounded dead-end depth with respect to every finite generating set. We also show that, in contrast, it has bounded retreat depth.