simplicial tree
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2017 ◽  
Vol 27 (01) ◽  
pp. 49-60
Author(s):  
A. Ivanov

We consider groups [Formula: see text] which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when [Formula: see text] acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.



2016 ◽  
Vol 53 ◽  
pp. 59-65 ◽  
Author(s):  
Woong Kook ◽  
Kang-Ju Lee
Keyword(s):  


2005 ◽  
Vol 15 (04) ◽  
pp. 765-798 ◽  
Author(s):  
DAN P. GURALNIK

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.



1997 ◽  
Vol 17 (4) ◽  
pp. 869-884 ◽  
Author(s):  
SA'AR HERSONSKY ◽  
JOHN HUBBARD

Let $T$ be a locally finite simplicial tree and let $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with $\Gamma$, which is also the Hausdorff dimension of the limit set of $\Gamma$; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of $\Gamma$. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its dimension; we give a new proof of this result. Finally, we show that the critical exponent is locally constant on the space of deformations of $\Gamma$.





1988 ◽  
Vol 103 (3) ◽  
pp. 409-426 ◽  
Author(s):  
Reinhard Diestel

The purpose of this paper is to give natural characterizations of the countable graphs that admit tree-decompositions or simplicial tree-decompositions into primes. Tree-decompositions were recently introduced by Robertson and Seymour in their series of papers on graph minors [7]. Simplicial tree-decompositions were first considered by Halin[6], being the most typical kind of ‘simplicial decomposition’ as introduced by Halin[5] in 1964. The problem of determining which infinite graphs admit a simplicial decomposition into primes has stood unresolved since then; a first solution for simplicial tree-decompositions was given in [2].



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