Title: LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS Author:Krön, Bernhard; Lehnert, Jörg; Seifter, Norbert; Teufl, Elmar Scale:
2015
Series:
20141222
Subject:20f65; 54e35; 20f18; 22e25; 05c63 Description:
Abstract We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.
Former:
20152014091222
Is part of:
Glasgow Mathematical Journal, 2015, Vol.57(3), pp.591-632
Arrangement:
201509
Identifier:
0017-0895 (ISSN); 1469-509X (E-ISSN); 10.1017/S0017089514000512 (DOI)
Title: Linear and projective boundaries in HNN-extensions and distortion phenomena Author:Krön, Bernhard; Lehnert, Jörg; Stein, Maya Subject:Mathematics - Group Theory ; 20f65 (Primary) 20e06, 05c63 (Secondary) Description:
Linear and projective boundaries of Cayley graphs were introduced in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits $g^\infty=\{g^i: i\in \mathbb N\}$, or orbits $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$, respectively, of non-torsion elements~$g$ of the group $G$, where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points $g^\infty$ and $g^{-\infty}$ in the linear boundary is bounded from below by $\sqrt{1/2}$, and we give an example of a group which has two antipodal elements of distance at most $\sqrt{12/17}
Identifier:
1210.4137 (ARXIV ID)
Title: Linear and projective boundaries in HNN-extensions and distortion phenomena Author:Krön, Bernhard; Lehnert, Jörg; Stein, Maya Contributor:Stein, Maya (pacrepositoryorg) Subject:Boundaries ; Orbits ; Lower Bounds ; Distortion ; Boundaries Description:
Linear and projective boundaries of Cayley graphs were introduced in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits \(g^\infty=\{g^i: i\in \mathbb N\}\), or orbits \(g^{\pm\infty}=\{g^i:i\in\mathbb Z\}\), respectively, of non-torsion elements~\(g\) of the group \(G\), where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points \(g^\infty\) and \(g^{-\infty}\) in the linear boundary is bounded from below by \(\sqrt{1/2}\), and we give an example of a group which has two antipodal elements of distance at most \(\sqrt{12/17}
Is part of:
arXiv.org, Aug 26, 2014
Identifier:
2331-8422 (E-ISSN)
Title: Linear and projective boundary of nilpotent groups Author:Krön, Bernhard; Lehnert, Jörg; Seifter, Norbert; Teufl, Elmar Subject:Mathematics - Group Theory ; 20f65 (Primary) 54e35, 20f18, 22e25, 05c63 (Secondary) Description:
We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub-(semi)-groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups. Comment: Version 2, 35 pages, 3 figures
Identifier:
1208.5405 (ARXIV ID)
Title: Linear and projective boundary of nilpotent groups Author:Krön, Bernhard; Lehnert, Jörg; Seifter, Norbert; Teufl, Elmar Contributor:Teufl, Elmar (pacrepositoryorg) Subject:Boundaries ; Metric Space ; Boundaries ; Random Walk ; Polynomials Description:
We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub-(semi)-groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth...
Is part of:
arXiv.org, Sep 20, 2013
Identifier:
2331-8422 (E-ISSN)
Title: Some remarks on depth of dead ends in groups Author:Lehnert, Jörg Subject:Mathematics - Group Theory ; Mathematics - Combinatorics ; 20f05 (Primary) 05c25 ; 20b30 (Secondary) Description:
It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to any set of generators. Partial results in this direction were obtained by \v{S}uni\'c and by Warshall. We improve these results by showing that abelian groups only have finitely many dead ends and that groups with more than one end (in the sense of Hopf and Freudenthal) have only dead ends of bounded depth. Only few examples of groups with unbounded dead end depth are known. We show that the Houghton group \Hou with respect to a standard generating set is a further example. In addition we introduce a stronger notion of depth of a dead end, called strong depth. The Houghton group \Hou has unbounded strong depth with respect to the same standard generating set. Comment: 10pages, version 2: corrected typos
Identifier:
math/0703636 (ARXIV ID)
Title: Some remarks on depth of dead ends in groups Author:Lehnert, Jörg Contributor:Lehnert, Jörg (pacrepositoryorg) Subject:Group Theory ; Generators ; Group Theory ; Combinatorics Description:
It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to any set of generators. Partial results in this direction were obtained by Šunić and by Warshall. We improve these results by showing that abelian groups only have finitely many dead ends and that groups with more than one end (in the sense of Hopf and Freudenthal) have only dead ends of bounded depth. Only few examples of groups with unbounded dead end depth are known. We show that the Houghton group \Hou with respect to a standard generating set is a further example. In addition we introduce a stronger notion of depth of a dead end, called strong depth. The Houghton group \Hou has unbounded strong depth with respect to the same standard generating set.
Is part of:
arXiv.org, Mar 22, 2007
Identifier:
2331-8422 (E-ISSN)
Title: Context-Freeness of Higman-Thompson group's co-word problem Author:Lehnert, Joerg; Schweitzer, Pascal Contributor:Schweitzer, Pascal (pacrepositoryorg) Subject:Permutations Description:
The co-word problem of a group G generated by a set X is defined as the set of words in X which do not represent 1 in G. We introduce a new method to decide if a permutation group has context-free co-word problem. We use this method to show, that the Higman-Thompson groups, and therefore the Houghton groups, have context-free co-word problem. We also give some examples of groups, that even have an easier co-word problem. We call this property semi-deterministic context-free. The second Houghton group belongs to this class.
Is part of:
arXiv.org, Jul 5, 2005
Identifier:
2331-8422 (E-ISSN)
Title: Context-Freeness of Higman-Thompson group's co-word problem Author:Lehnert, Joerg; Schweitzer, Pascal Subject:Mathematics - Group Theory ; 20f10 ; 68q45 (Primary) 03d40 (Secondary) Description:
The co-word problem of a group G generated by a set X is defined as the set of words in X which do not represent 1 in G. We introduce a new method to decide if a permutation group has context-free co-word problem. We use this method to show, that the Higman-Thompson groups, and therefore the Houghton groups, have context-free co-word problem. We also give some examples of groups, that even have an easier co-word problem. We call this property semi-deterministic context-free. The second Houghton group belongs to this class. Comment: 9 pages
Identifier:
math/0507090 (ARXIV ID)