scholarly journals On the group of spheromorphisms of a homogeneous non-locally finite tree

2020 ◽  
Vol 84 (6) ◽  
pp. 1161-1191
Author(s):  
Yu. A. Neretin
Keyword(s):  
Finite Tree ◽  
Locally Finite

scholarly journals Distinguishability of Locally Finite Trees

10.37236/947 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Mark E. Watkins ◽  
Xiangqian Zhou
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Inverse Image ◽  
Vertex Coloring ◽  
Finite Tree ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Nonidentity Element ◽  
Locally Countable ◽  
Proper Vertex

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

scholarly journals Non-minimal tree actions and the existence of non-uniform tree lattices

2004 ◽  
Vol 70 (2) ◽  
pp. 257-266
Author(s):  
Lisa Carbone
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Connected Graph ◽  
Locally Compact Group ◽  
Minimal Tree ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Finite Tree ◽  
Locally Finite ◽  
Uniform Tree

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

CONJUGATION IN TREE AUTOMORPHISM GROUPS

2001 ◽  
Vol 11 (05) ◽  
pp. 529-547 ◽  
Author(s):  
PIOTR W. GAWRON ◽  
VOLODYMYR V. NEKRASHEVYCH ◽  
VITALY I. SUSHCHANSKY
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Rooted Trees ◽  
Full Description ◽  
Finite Tree ◽  
Locally Finite ◽  
Infinite Sequences

It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

scholarly journals CHABAUTY LIMITS OF SIMPLE GROUPS ACTING ON TREES

2018 ◽  
Vol 19 (4) ◽  
pp. 1093-1120
Author(s):  
Pierre-Emmanuel Caprace ◽  
Nicolas Radu
Keyword(s):  
Finite Index ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Simple Groups ◽  
Finite Tree ◽  
Hausdorff Topology ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Bounded Number ◽  
Groups Acting On Trees

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

scholarly journals CONTEXT-FREE GROUPS AND THEIR STRUCTURE TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 611-642 ◽  
Author(s):  
VOLKER DIEKERT ◽  
ARMIN WEIß
Keyword(s):  
Free Groups ◽  
Finite Graph ◽  
Fundamental Result ◽  
Direct Construction ◽  
Finite Tree ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Structure Tree ◽  
Tree Width ◽  
Context Free

Let Γ be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Γ. Once the tree is constructed, Bass–Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as finitely generated (f.g.) virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained. We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.

scholarly journals Effective mixing and counting in Bruhat–Tits trees

2016 ◽  
Vol 38 (1) ◽  
pp. 257-283
Author(s):  
SANGHOON KWON
Keyword(s):  
Algebraic Group ◽  
Spectral Gap ◽  
Discrete Subgroup ◽  
Length Spectrum ◽  
Finite Tree ◽  
Locally Finite ◽  
Exponential Mixing ◽  
Invariant Potential ◽  
Mixing Property

Let ${\mathcal{T}}$ be a locally finite tree, $\unicode[STIX]{x1D6E4}$ be a discrete subgroup of $\,\operatorname{Aut}\,({\mathcal{T}})$ and $\widetilde{F}$ be a $\unicode[STIX]{x1D6E4}$-invariant potential. Suppose that the length spectrum of $\unicode[STIX]{x1D6E4}$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}\rightarrow \unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}$ with respect to the measure $m_{\unicode[STIX]{x1D6E4},F}^{\unicode[STIX]{x1D708}^{-},\unicode[STIX]{x1D708}^{+}}$ under the assumption that $\unicode[STIX]{x1D6E4}$ is full and $(\unicode[STIX]{x1D6E4},\widetilde{F})$ has a weighted spectral gap. We also obtain the effective formula for the number of $\unicode[STIX]{x1D6E4}$-orbits with weights in a Bruhat–Tits tree ${\mathcal{T}}$ of an algebraic group.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

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