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Isoperimetric and isodiametric functions of groups
 Ann. of Math
"... This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2Cm for some constant C> 0 then n α is eq ..."
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Cited by 33 (15 self)
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This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2Cm for some constant C> 0 then n α is equivalent (“big O”) to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n 3/4α. On the other hand if n α is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C2 22Cm for some constant C. This implies that, say, functions n π+1, n e2 and n α for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that n π and n α for all rational numbers α ≥ 3 are equivalent to the smallest isodiametric functions of finitely presented groups.
Lpresentations and branch groups
, 2002
"... Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1. ..."
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Cited by 24 (8 self)
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Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1.
Isoperimetric functions of groups and computational complexity of the word problem
 Annals of Mathematics (accepted). Mathematics arXiv, math.GR/9811106, http://front.math.ucdavis.edu
, 1998
"... \Lambda ..."
Endomorphic presentations of branch groups
 J.Algebra
"... Abstract. We introduce “endomorphic presentations”, or Lpresentations: group presentations whose relations are iterated under a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations, generalisin ..."
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Cited by 11 (3 self)
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Abstract. We introduce “endomorphic presentations”, or Lpresentations: group presentations whose relations are iterated under a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations, generalising results by Igor Lysionok and Said Sidki. 1.
Decision problems in group theory
 Proc. London Math. Soc
, 1982
"... At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an ex ..."
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Cited by 5 (0 self)
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At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular
Algorithmic and asymptotic properties of groups
 the Proceedings of ICM in
, 2006
"... Abstract. This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters). ..."
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Cited by 1 (0 self)
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Abstract. This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters).