Compact groups with many elements of bounded order

Locally compact groups with every isometric action bounded or proper

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, inc…

Elementary subgroups of relatively hyperbolic groups and bounded generation

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_λ,λ\in Λ\} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_λ,λ\in Λ\} \cup \{Q\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_λ$, $λ\in Λ$, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementar…

Bounded Engel elements in residually finite groups

Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a non-commutator word. Assume that $G$ is a $d$-generator residually finite group in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$ has $\{d,n,w\}$-bounded nilpotency class.

Bounded generation by root elements for Chevalley groups defined over rings of integers of function fields with an application in strong boundedness

Bounded generation by root elements is a property which has been widely studied for various types of linear algebraic groups defined over rings of integers in algebraic number fields. However, when considering global function fields, there are not many results beyond the treatment of special cases due to Nica and Queen. In this paper, we use model theoretic methods due to Carter, Keller and Paige written up by Morris to prove bounded generation by root elements for simply connected, split Chevalley groups defined over the ring of all integers in a global function field. We further apply this b…

Elements of uniformly bounded word-length in groups

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and finiteness of $G_{bound}$. We prove that if $G$ is virtually abelian then $G_{bound}$ is finite. In contrast with numerous examples where $G_{bound}$ is trivial, we show that for every finite group $A$, there exists an infinite group $G$ with $G_{bound}=A$. This group $G$ can be chosen among torsion groups. We also study the group $G_{bound}(d)$ of elements with …

Bounded Conjugators For Real Hyperbolic and Unipotent Elements in Semisimple Lie Groups

Let $G$ be a real semisimple Lie group with trivial centre and no compact factors. Given a conjugate pair of either real hyperbolic elements or unipotent elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that $d_G(1,g) \leq L(d_G(1,u)+d_G(1,v))$, where $L$ is a positive constant which will depend on some property of $a$ and $b$. For the vast majority of such elements however, $L$ can be assumed to be a uniform constant.

Elementary abelian 2 subgroups of compact Lie groups

We classify elementary abelian 2 subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian $p$ subgroups of compact (or linear algebraic) simple groups of adjoint type.

Chevalley 群的均匀有界初等生成

Uniform bounded elementary generation of Chevalley groups

本文证明秩至少为2的Chevalley群在任意算术型Dedekind环上具有均匀有界初等生成，给出了普适的宽度上界。

平面Cremona群中代数元素的有限生成子群是有界的

Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded

该研究证明平面Cremona群中仅含代数元素的有限生成子群次数有界，基于无穷直和上“良好”作用的更一般结论，还刻画了这类子群的次数增长特性。

The Engel elements in generalized FC-groups

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements con…

Fixed points for bounded orbits in Hilbert spaces

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact groups (e.g. countable groups). Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau--Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for…

Amenable actions, invariant means and bounded cohomology

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-Čech compactification of the group we obtain a cohomologic…