scholarly journals Distinguishing Numbers for Graphs and Groups

10.37236/1816 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Julianna Tymoczko
Keyword(s):  
Group Action ◽  
General Problem ◽  
Group Actions ◽  
Graph Automorphism ◽  
Arbitrary Group ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Trivial Graph ◽  
Open Question

A graph $G$ is distinguished if its vertices are labelled by a map $\phi: V(G) \longrightarrow \{1,2,\ldots, k\}$ so that no non-trivial graph automorphism preserves $\phi$. The distinguishing number of $G$ is the minimum number $k$ necessary for $\phi$ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of $\Gamma$ on a set $X$. A labelling $\phi: X \longrightarrow \{1,2,\ldots,k\}$ is distinguishing if no element of $\Gamma$ preserves $\phi$ except those which fix each element of $X$. The distinguishing number of the group action on $X$ is the minimum $k$ needed for $\phi$ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of $S_n$ on a set with distinguishing number $n$, answering an open question of Albertson and Collins.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

2014 ◽  
Vol 23 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Toru Ikeda
Keyword(s):  
Finite Group ◽  
Group Action ◽  
Group Actions ◽  
Infinite Family ◽  
Finite Group Action ◽  
Finite Group Actions ◽  
Spatial Graph ◽  
Cell Decompositions ◽  
Spatial Graphs ◽  
Ideal Triangulations

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

scholarly journals An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

scholarly journals Sofic entropy and amenable groups

2011 ◽  
Vol 32 (2) ◽  
pp. 427-466 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Group Action ◽  
Amenable Group ◽  
Group Actions ◽  
Amenable Groups ◽  
Orbit Equivalent ◽  
Sofic Entropy ◽  
Classical Entropy

AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

scholarly journals Rigid group actions on complex tori are projective (after Ekedahl)

2020 ◽  
Vol 22 (07) ◽  
pp. 1950092 ◽  
Author(s):  
Fabrizio Catanese ◽  
Andreas Demleitner
Keyword(s):  
Group Action ◽  
Group Actions ◽  
Detailed Proof ◽  
Complex Tori ◽  
Rigid Group

In this paper, we give a detailed proof of a result due to Torsten Ekedahl, describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones.

Group Actions on Quasi-Baer Rings

2009 ◽  
Vol 52 (4) ◽  
pp. 564-582 ◽  
Author(s):  
Hai Lan Jin ◽  
Jaekyung Doh ◽  
Jae Keol Park
Keyword(s):  
Finite Group ◽  
Group Action ◽  
Semiprime Ring ◽  
Group Actions ◽  
Group Rings ◽  
Finite Group Action ◽  
Baer Rings ◽  
Skew Group Rings ◽  
A Finite Group ◽  
The Right

AbstractA ring R is called quasi-Baer if the right annihilator of every right ideal of R is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to C*-algebras. Various examples to illustrate and delimit our results are provided.

scholarly journals Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

1981 ◽  
Vol 1 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Klaus Schmidt
Keyword(s):  
Group Action ◽  
Measure Space ◽  
Group Actions ◽  
Finite Measure ◽  
Countable Group ◽  
Invariant Sets ◽  
Amenable Groups ◽  
Strong Ergodicity ◽  
Property T ◽  
Invariant Means

AbstractThis paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.

scholarly journals On the Minimum Number of Monochromatic Generalized Schur Triples

10.37236/6490 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Thotsaporn Thanatipanonda ◽  
Elaine Wong
Keyword(s):  
Positive Integer ◽  
General Problem ◽  
Original Proof ◽  
Minimum Number ◽  
Fixed Positive Integer

The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.

scholarly journals Novikov-Shubin invariants for arbitrary group actions and their positivity

1999 ◽  
pp. 159-176 ◽  
Author(s):  
Wolfgang Lück ◽  
Holger Reich ◽  
Thomas Schick
Keyword(s):  
Group Actions ◽  
Arbitrary Group

scholarly journals The Complexity of Computing the Cylindrical and the $t$-Circle Crossing Number of a Graph

10.37236/7193 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Frank Duque ◽  
Hernán González-Aguilar ◽  
César Hernández-Vélez ◽  
Jesús Leaños ◽  
Carolina Medina
Keyword(s):  
Analogous Result ◽  
Natural Generalization ◽  
Crossing Number ◽  
Number Problem ◽  
Minimum Number ◽  
Np Complete ◽  
Concentric Circles ◽  
Definition Of ◽  
Open Question

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper, we prove that the problem of deciding whether a given graph admits a cylindrical embedding is NP-complete, and as a consequence we show that the \(t\)-cylindrical crossing number problem is also NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number,  namely the \(t\)-crossing number.

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