THE LINEARITY OF THE CONJUGACY PROBLEM IN WORD-HYPERBOLIC GROUPS

2006 ◽  
Vol 16 (02) ◽  
pp. 287-305 ◽  
Author(s):  
DAVID EPSTEIN ◽  
DEREK HOLT
Keyword(s):  
Normal Form ◽  
Theoretical Result ◽  
Linear Time ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Constant Time ◽  
Word Hyperbolic Group ◽  
Group A ◽  
Word Hyperbolic Groups

The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in which basic arithmetical operations on integers are assumed to take place in constant time. The constants involved in the linear time solution are all computable explicitly. We also give a proof of the result of Mike Shapiro that in a word-hyperbolic group a word in the generators can be transformed into short-lex normal form in linear time. This is used in the proof of our main theorem, but is a significant theoretical result of independent interest, which deserves to be in the literature. Previously the best known result was a quadratic estimate.

Amalgamated Products and the Howson Property

1997 ◽  
Vol 40 (3) ◽  
pp. 330-340 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Amalgamated Products ◽  
Word Hyperbolic Groups ◽  
Free Word ◽  
Torsion Free ◽  
Maximal Cyclic Subgroup

AbstractWe show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

scholarly journals Quasiconvexity and Amalgams

1997 ◽  
Vol 07 (06) ◽  
pp. 771-811 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Cyclic Subgroup ◽  
Hyperbolic Group ◽  
Hyperbolic Surface ◽  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
3 Dimensional ◽  
Word Hyperbolic Group ◽  
Amalgamated Free Product ◽  
One Relator Groups ◽  
Word Hyperbolic Groups

We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian groups have property (Q). We also give some applications of our results to one-relator groups and exponential groups.

scholarly journals THE CONJUGACY PROBLEM IN HYPERBOLIC GROUPS FOR FINITE LISTS OF GROUP ELEMENTS

2013 ◽  
Vol 23 (05) ◽  
pp. 1127-1150 ◽  
Author(s):  
D. J. BUCKLEY ◽  
DEREK F. HOLT
Keyword(s):  
Upper Bound ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Generating Set ◽  
Running Time ◽  
Word Hyperbolic Group ◽  
Practical Algorithm

Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. An (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element, should one exist. The algorithm runs in time O(mμ), where μ is an upper bound on the length of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centralizer of A, with the same bound on running time.

scholarly journals Quasiregular self-mappings of manifolds and word hyperbolic groups

2007 ◽  
Vol 143 (6) ◽  
pp. 1613-1622 ◽  
Author(s):  
Martin Bridson ◽  
Aimo Hinkkanen ◽  
Gaven Martin
Keyword(s):  
Hyperbolic Group ◽  
Quasiregular Mapping ◽  
Hyperbolic Groups ◽  
Quasiregular Mappings ◽  
Word Hyperbolic Group ◽  
Word Hyperbolic Groups ◽  
Mostow Rigidity ◽  
Negative Sectional Curvature ◽  
Free Word ◽  
Torsion Free

AbstractAn extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature ($n\neq 4$), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups

1996 ◽  
Vol 48 (6) ◽  
pp. 1224-1244 ◽  
Author(s):  
Ilya Kapovich ◽  
Hamish Short
Keyword(s):  
Finite Index ◽  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Quasiconvex Subgroups ◽  
Word Hyperbolic Groups ◽  
Finite Index Subgroup

AbstractAnalogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

scholarly journals Time complexity of the conjugacy problem in relatively hyperbolic groups

2015 ◽  
Vol 25 (05) ◽  
pp. 689-723 ◽  
Author(s):  
Inna Bumagin
Keyword(s):  
Polynomial Time ◽  
Positive Curvature ◽  
Nauk Sssr ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Search Problem ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

If u and v are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for u and v can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok in [Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53(4) (1989) 814–832, 912]. Bridson and Haefliger showed in [Metrics Spaces of Non-Positive Curvature (Springer-Verlag, Berlin, 1999)] that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if u and v are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for u and v is linear in terms of their lengths.

scholarly journals SOLVING THE WORD PROBLEM IN REAL TIME

2001 ◽  
Vol 63 (3) ◽  
pp. 623-639 ◽  
Author(s):  
DEREK F. HOLT ◽  
SARAH REES
Keyword(s):  
Recent Result ◽  
Real Time ◽  
Word Problem ◽  
Turing Machine ◽  
Linear Time ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Turing Machines ◽  
Geometrically Finite ◽  
Word Hyperbolic Groups

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.

CONJUGACY OF FINITE SUBSETS IN HYPERBOLIC GROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 725-756 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JAMES HOWIE
Keyword(s):  
Linear Function ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Time Algorithm ◽  
Hyperbolic Groups ◽  
Finitely Presented Groups ◽  
Curved Spaces ◽  
Negatively Curved ◽  
Finitely Presented ◽  
Torsion Free

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x-1 Ax = B for some x ∈ Γ with ǁxǁ less than a linear function of max {ǁγǁ : γ ∈ A ∪ B}. (The coefficients of this linear function depend only on k and δ.) These results have implications for group-based cryptography and the geometry of homotopies in negatively curved spaces. In an appendix, we give examples of finitely presented groups in which the conjugacy problem for elements is soluble but the conjugacy problem for finite lists is not.

scholarly journals Combable functions, quasimorphisms, and the central limit theorem

2009 ◽  
Vol 30 (5) ◽  
pp. 1343-1369 ◽  
Author(s):  
DANNY CALEGARI ◽  
KOJI FUJIWARA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Word Length ◽  
Hyperbolic Groups ◽  
Finite State Automaton ◽  
List Type ◽  
Generating Sets ◽  
Finite State ◽  
Word Hyperbolic Groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

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