Hyperbolic hydra

Brady, Noel; Dison, Will; Riley, Tim

doi:10.4171/ggd/212

Cited by 8 publications

(20 citation statements)

References 13 publications

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“…In the second step the prefix a is omitted, and one gets from babab the word babbabb via ρ 1 and so on. The following diagram summarizes the battle we are describing: S k e t c h o f t h e p r o o f. We give just an idea of the proof with a computational argument (for the proof see [6,12]), since the general case follows analogously. Assume…”

Section: 1])mentioning

confidence: 99%

“…, which is known in number theory as Ackermann function (see [6,12]). Among its properties, we have that H k (n) ≥ A k (n), for all k ≥ 3 and n > 0.…”

Section: S K E T C H O F T H E P R O O F Given K N > 0 We May Defimentioning

confidence: 99%

“…Ò Ø ÓÒ 1.2º (See [6,12]) Let A = {a, b} be an alphabet on 2 letters, w ∈ A * a word of length > 1 and ρ 1 the function, from A * to A * , defined by…”

Section: 1])mentioning

confidence: 99%

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Some considerations on Hydra groups and a new bound for the length of words

Otera

Russo

2014

Mathematica Slovaca

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ABSTRACT. After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins. Hydra groupsThe combinatorics of words is increased significatively in the last years, since it is a powerful tool for the description of several processes in pure and applied sciences. A classic reference on the topic is [24] and will be used for the basic notions. We recall that a nonempty set A (which will be always assumed to be finite) is said to be an alphabet and its elements are called letters. By A + we denote the free semigroup generated by A, i.e. the set of all finite sequences of letters with the operation of concatenation. Elements of A + will be called words. It is well known that each word can be written in a unique way as a sequence of letters, so it is possible to define for each word w its length |w|:It is useful (though not necessary) to introduce the formal empty word ε: that is a word of length 0 such that2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 57M50; Secondary 68R05, 68Q15. K e y w o r d s: isoperimetric functions, finitely presented groups, lengths of words, counterpoint, Myhill-Nerode's theorem.

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Section: 1])mentioning

confidence: 99%

“…, which is known in number theory as Ackermann function (see [6,12]). Among its properties, we have that H k (n) ≥ A k (n), for all k ≥ 3 and n > 0.…”

Section: S K E T C H O F T H E P R O O F Given K N > 0 We May Defimentioning

confidence: 99%

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Some considerations on Hydra groups and a new bound for the length of words

Otera

Russo

2014

Mathematica Slovaca

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“…One can also try to use the hydra groups [10,14] to construct HNN extensions as above with Dehn functions bigger than any prescribed Ackermann function. The question of whether these groups are residually finite was open when the first version of this paper was written, and is now answered in negative in [49].…”

Section: Lemma 11 If the Group T Is Residually Finite Then H Is Closmentioning

confidence: 99%

Algorithmically complex residually finite groups

2017

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“…In [1] and [2] the authors construct hyperbolic semidirect products N ⋊ Z with N free, such that the distortion of N is superpolynomial. Such extensions do not fall into the scope of Jolissaint's theorem, while they still satisfy assumptions of Theorem 4.1.…”

Section: Final Remarksmentioning

confidence: 99%

Property of rapid decay for extensions of compactly generated groups

Garncarek

2015

Publicacions Matemàtiques

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ABSTRACT. In the article we settle down the problem of permanence of property RD under group extensions. We show that if 1 → N → G → Q → 1 is a short exact sequence of compactly generated groups such that Q has property RD, and N has property RD with respect to the restriction of a word-length on G, then G has property RD.We also generalize the result of Ji and Schweitzer stating that locally compact groups with property RD are unimodular. Namely, we show that any automorphism of a locally compact group with property RD which distorts distances subexponentially, preserves the Haar measure.

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Hyperbolic hydra

Abstract: We give examples of hyperbolic groups with finite-rank free subgroups of huge (Ackermannian) distortion.

Cited by 8 publications

References 13 publications

Some considerations on Hydra groups and a new bound for the length of words

Some considerations on Hydra groups and a new bound for the length of words

Algorithmically complex residually finite groups

Property of rapid decay for extensions of compactly generated groups

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