Remarks on the actions of some groups on the product of Hadamard spaces

Zunwu He

Home/Archive/arXiv

arXiv · 2009.13800

Remarks on the actions of some groups on the product of Hadamard spaces

Zunwu He

View original source ↗Read source document ↗Export metadata ↓

Abstract

For a product of Hadamard spaces $X=X_1\times X_2$ on which some group $G\subset Is(X_1)\times Is(X_2)$ acting, G. Link arXiv:1107.3755v1 introduced the growth rate $δ_θ$ of slope $θ$ to construct a $G-$invariant $(b,θ)$-density. First, we show that $δ_θ$ is continuous in the slope $θ$ as the above group action with some mild condition. Second, we give a negative answer to a question raised by G. Link in \cite{1} in general. And further results about the question are discussed. In the end, we can extent the $(b,θ)$-densities in some reasonable condition.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zunwu He. 2020-12-11. Remarks on the actions of some groups on the product of Hadamard spaces. https://arxiv.org/abs/2009.13800

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On commensurators of free groups and free pro-p groups

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

math.GR↗

arXiv2026

Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,λ\right)=\mathcal{D}_{2}\left(G,ρ\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.

math.GR↗

arXiv2026

Quandles from group actions and a Cayley-type embedding theorem

A quandle is an algebraic system that can be regarded as a generalization of the conjugation operation in groups. We study a quandle construction associated with group actions and determine its structural properties, including its inner automorphism group, connected components, and subquandles. As a principal application, we establish a Cayley-type embedding theorem for finite quandles. Applying the construction to the natural action of the symmetric group, we obtain, for each $n$, a single quandle into which every quandle of cardinality $n$ embeds.

math.GR↗