The structure of hyperbolic spaces at infinity

Dyubina, Anna; Polterovich, Iosif

doi:10.1070/rm1998v053n05abeh000080

Cited by 6 publications

(1 citation statement)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 10 we discuss some models of rbRT r that have been constructed within the theory of R-trees by Dyubina and Polterovich [12,13] and some other models that arise in geometric group theory.…”

Section: S Carlisle and C W Hensonmentioning

confidence: 99%

Model theory of R-trees

Carlisle

Henson

2020

J. Log. Anal.

View full text Add to dashboard Cite

We show the theory of pointed $\R$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\R$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

show abstract

Section: S Carlisle and C W Hensonmentioning

confidence: 99%

Model theory of R-trees

Carlisle

Henson

2020

J. Log. Anal.

View full text Add to dashboard Cite

show abstract

On the geometry of infinite cyclic subgroups

Erschler-Dyubina

2002

Isr. J. Math.

View full text Add to dashboard Cite

Infinite Strings and Their Large Scale Properties

Khoussainov¹,

Takisaka²

2020

J. symb. log.

View full text Add to dashboard Cite

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string has weaker large scale geometry than that of if there is color preserving bi-Lipschitz map from into with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness.

show abstract

The structure of hyperbolic spaces at infinity

Cited by 6 publications

References 2 publications

Model theory of R-trees

Model theory of R-trees

On the geometry of infinite cyclic subgroups

Infinite Strings and Their Large Scale Properties

Contact Info

Product

Resources

About