Dehn Functions and l 1-norms of Finite Presentations

1992 ◽  
pp. 195-224 ◽  
Author(s):  
S. M. Gersten
Keyword(s):  
Dehn Functions ◽  
Finite Presentations

scholarly journals ON THE GROWTH OF GROUPS AND AUTOMORPHISMS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 869-874 ◽  
Author(s):  
MARTIN R. BRIDSON
Keyword(s):  
Finitely Generated ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Growth Functions ◽  
Growth Of Groups

We consider the growth functions βΓ(n) of amalgamated free products Γ = A *C B, where A ≅ B are finitely generated, C is free abelian and |A/C| = |A/B| = 2. For every d ∈ ℕ there exist examples with βΓ(n) ≃ nd+1βA(n). There also exist examples with βΓ(n) ≃ en. Similar behavior is exhibited among Dehn functions.

scholarly journals Pro-finite Presentations

2001 ◽  
Vol 242 (2) ◽  
pp. 672-690 ◽  
Author(s):  
Alexander Lubotzky
Keyword(s):  
Finite Presentations

Finiteness and Dehn functions of automatic monoids having directed fellow traveller property

2009 ◽  
Vol 79 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Xiaofeng Wang ◽  
Wanwen Xie ◽  
Hanling Lin
Keyword(s):  
Dehn Functions ◽  
Fellow Traveller ◽  
Fellow Traveller Property

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.

Second Order Dehn Functions of Groups

1998 ◽  
Vol 49 (1) ◽  
pp. 1-30 ◽  
Author(s):  
J. M. Alonso ◽  
W. A. Bogley ◽  
R. M. Burton ◽  
S. J. Pride ◽  
X. Wang
Keyword(s):  
Second Order ◽  
Dehn Functions

POLYNOMIAL DEHN FUNCTIONS AND THE LENGTH OF ASYNCHRONOUSLY AUTOMATIC STRUCTURES

2002 ◽  
Vol 85 (2) ◽  
pp. 441-466 ◽  
Author(s):  
MARTIN R. BRIDSON
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Dehn Function ◽  
Arbitrary Degree ◽  
Dehn Functions ◽  
Automatic Structures ◽  
Automatic Groups ◽  
Observed Behaviour ◽  
Exact Calculations ◽  
Do So

We extend the range of observed behaviour among length functions of optimal asynchronously automatic structures. We do so by means of a construction that yields asynchronously automatic groups with finite aspherical presentations where the Dehn function of the group is polynomial of arbitrary degree. Many of these groups can be embedded in the automorphism group of a free group. Moreover, the fact that the groups have aspherical presentations makes them useful tools in the search to determine the spectrum of exponents for second order Dehn functions. We contribute to this search by giving the first exact calculations of groups with quadratic and superquadratic exponents. 2000 Mathematical Subject Classification:20F06, 20F65, 20F69.

scholarly journals Hydra group doubles are not residually finite

2016 ◽  
Vol 8 (2) ◽  
Author(s):  
Kristen Pueschel
Keyword(s):  
Finite Groups ◽  
Recursive Function ◽  
Dehn Functions ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Finitely Presented

AbstractIn 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function

scholarly journals Generalized small cancellation presentations for automatic groups

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert H. Gilman
Keyword(s):  
Small Cancellation ◽  
Automatic Groups ◽  
Finite Presentations

AbstractBy a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.

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.

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