Taming the hydra: The word problem and extreme integer compression

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.

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Embeddings into groups with only a few defining relations

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700019066 ◽

1974 ◽

Vol 18 (1) ◽

pp. 1-7 ◽

Cited By ~ 3

Author(s):

W. W. Boone ◽

D. J. Collins

Keyword(s):

Word Problem ◽

Finitely Presented Groups ◽

Fixed Group ◽

Defining Relations ◽

Finitely Presented Group ◽

One Relator Groups ◽

Finitely Presented ◽

Unsolvable Word Problem ◽

Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

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Annular Dehn functions of groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032433 ◽

1998 ◽

Vol 58 (3) ◽

pp. 453-464 ◽

Cited By ~ 1

Author(s):

Stephen G. Brick ◽

Jon M. Corson

Keyword(s):

Conjugacy Problem ◽

Upper Bounds ◽

Dehn Function ◽

Free Products ◽

Dehn Functions ◽

Finitely Presented Group ◽

Hnn Extensions ◽

Finite Subgroups ◽

Boundary Curves ◽

Finitely Presented

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

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GROUPS WITH QUADRATIC-NON-QUADRATIC DEHN FUNCTIONS

International Journal of Algebra and Computation ◽

10.1142/s0218196707003688 ◽

2007 ◽

Vol 17 (02) ◽

pp. 401-419 ◽

Cited By ~ 4

Author(s):

ALEXANDER YU. OL'SHANSKII

Keyword(s):

Quadratic Function ◽

Dehn Function ◽

Dehn Functions ◽

Finitely Presented Group ◽

Finitely Presented

We construct a finitely presented group G with non-quadratic Dehn function f majorizable by a quadratic function on arbitrary long intervals.

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Dehn Functions

Office Hours with a Geometric Group Theorist ◽

10.23943/princeton/9780691158662.003.0008 ◽

2017 ◽

Author(s):

Timothy Riley

Keyword(s):

Word Problem ◽

Riemannian Manifolds ◽

Isoperimetric Problem ◽

Complexity Measure ◽

Research Projects ◽

Dehn Function ◽

Continuous Version ◽

Dehn Functions ◽

Jigsaw Puzzles ◽

Combinatorial Space

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a complexity measure for the word problem and shows that, for a word w, the problem of finding a sequence of free reductions, free expansions, and applications of defining relators that carries it to the empty word is equivalent to solving the puzzle where, starting from some vertex υ‎, one reads w around the initial circle of rods. The chapter also explains how the Dehn function corresponds to an isoperimetric problem in a combinatorial space, the Cayley 2-complex, and describes a continuous version of this, via group actions, along with the isoperimetry in Riemannian manifolds. Finally, it defines the Dehn function as a quasi-isometry invariant. The discussion includes exercises and research projects.

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FINITELY PRESENTED GROUP WHOSE WORD PROBLEM HAS THE SAME DEGREE AS THAT OF AN ARBITRARILY GIVEN THUE SYSTEM (AN APPLICATION OF METHODS OF BRITTON)

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.53.2.265 ◽

1965 ◽

Vol 53 (2) ◽

pp. 265-269 ◽

Cited By ~ 7

Author(s):

W. W. Boone

Keyword(s):

Word Problem ◽

Finitely Presented Group ◽

Thue System ◽

Finitely Presented

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Dehn functions of finitely presented metabelian groups

Journal of Group Theory ◽

10.1515/jgth-2020-0182 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wenhao Wang

Keyword(s):

Abelian Group ◽

Upper Bound ◽

Finite Rank ◽

Free Abelian Group ◽

Metabelian Group ◽

Finitely Generated ◽

Dehn Function ◽

Metabelian Groups ◽

Dehn Functions ◽

Finitely Presented

Abstract In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

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Conjugacy problem in groups with quadratic Dehn function

Bulletin of Mathematical Sciences ◽

10.1142/s1664360719500231 ◽

2020 ◽

Vol 10 (01) ◽

pp. 1950023 ◽

Cited By ~ 1

Author(s):

A. Yu. Olshanskii ◽

M. V. Sapir

Keyword(s):

Conjugacy Problem ◽

Dehn Function ◽

Finitely Presented Group ◽

Finitely Presented

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

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THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

International Journal of Algebra and Computation ◽

10.1142/s0218196704001980 ◽

2004 ◽

Vol 14 (05n06) ◽

pp. 569-626 ◽

Cited By ~ 30

Author(s):

JEAN-CAMILLE BIRGET

Keyword(s):

Word Problem ◽

Simple Groups ◽

Direct Products ◽

Turing Machines ◽

Group V ◽

Dehn Functions ◽

Finitely Presented Groups ◽

Distortion Functions ◽

The 1960S ◽

Finitely Presented

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

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