scholarly journals Annular Dehn functions of groups

1998 ◽  
Vol 58 (3) ◽  
pp. 453-464 ◽  
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.

scholarly journals Taming the hydra: The word problem and extreme integer compression

2018 ◽  
Vol 28 (07) ◽  
pp. 1299-1381
Author(s):  
W. Dison ◽  
E. Einstein ◽  
T. R. Riley
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Complexity Measure ◽  
Dehn Function ◽  
Dehn Functions ◽  
Fast Growing ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Direct Attack ◽  
Finitely Presented

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.

scholarly journals Conjugacy problem in groups with quadratic Dehn function

2020 ◽  
Vol 10 (01) ◽  
pp. 1950023 ◽  
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.

scholarly journals GROUPS WITH QUADRATIC-NON-QUADRATIC DEHN FUNCTIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 401-419 ◽  
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.

Second order Dehn functions of Pride groups

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Peter Davidson
Keyword(s):  
Upper Bounds ◽  
Second Order ◽  
Artin Group ◽  
Dehn Function ◽  
Dehn Functions

Abstract.Under suitable conditions upper bounds of second order Dehn functions of Pride groups are obtained. From this we show that the second order Dehn function of a right-angled Artin group is at most quadratic.

Dehn functions of finitely presented metabelian groups

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.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

2009 ◽  
Vol 02 (04) ◽  
pp. 611-635 ◽  
Author(s):  
K. Kalorkoti
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Computation Step ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Solvable Word ◽  
Extended Notion ◽  
Ordered Pairs

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

scholarly journals Ascending HNN-extensions and properly 3-realisable groups

2005 ◽  
Vol 72 (2) ◽  
pp. 187-196 ◽  
Author(s):  
Francisco F. Lasheras
Keyword(s):  
Homotopy Type ◽  
Universal Cover ◽  
Finitely Presented Group ◽  
Manifold With Boundary ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Proper Homotopy ◽  
Finitely Presented

In this paper, we show that any ascending HNN-extension of a finitely presented group is properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1(K) ≅ G and whose universal cover K̃ has the proper homotopy type of a (PL) 3-manifold (with boundary).

scholarly journals A parallel evolutionary approach to solving systems of equations in polycyclic groups

2016 ◽  
Vol 8 (2) ◽  
Author(s):  
Matthew J. Craven ◽  
Daniel Robertz
Keyword(s):  
Linear Algebra ◽  
High Performance ◽  
Conjugacy Problem ◽  
Evolutionary Approach ◽  
Alternative Measure ◽  
Systems Of Equations ◽  
Key Exchange Protocol ◽  
Finitely Presented Group ◽  
Polycyclic Groups ◽  
Finitely Presented

AbstractThe Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced.

Unsolvable Problems in Groups With Solvable Word Problem

1970 ◽  
Vol 22 (4) ◽  
pp. 836-838 ◽  
Author(s):  
James McCool
Keyword(s):  
Word Problem ◽  
Conjugacy Problem ◽  
The Other ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
Other Hand ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Effective Procedures ◽  
Solvable Word

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

scholarly journals The fixed point theorem for simplicial nonpositive curvature

2008 ◽  
Vol 144 (3) ◽  
pp. 683-695
Author(s):  
PIOTR PRZYTYCKI
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Nonpositive Curvature ◽  
Free Products ◽  
Locally Finite ◽  
Hnn Extensions ◽  
Finite Subgroups ◽  
The Fixed Point Theorem ◽  
A Finite Group ◽  
Free Products With Amalgamation

AbstractWe prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.

Sign in / Sign up

Export Citation Format

Share Document