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 From automatic structures to automatic groups

2014 ◽  
Vol 8 (1) ◽  
pp. 157-198 ◽  
Author(s):  
Olga Kharlampovich ◽  
Bakhadyr Khoussainov ◽  
Alexei Miasnikov
Keyword(s):  
Automatic Structures ◽  
Automatic Groups

scholarly journals Does the intensity of prudential regulation affect banks? Evidence from the 2007-2009 crises.

2012 ◽  
Vol 1 (3) ◽  
pp. 14-26 ◽  
Author(s):  
Isabel Argimon ◽  
Gerard Arqué Castells ◽  
Francesc Rodríguez Tous
Keyword(s):  
Capital Requirements ◽  
Capital Regulation ◽  
Risk Sensitivity ◽  
The European Union ◽  
Regulatory Capital ◽  
European Banks ◽  
Observed Behaviour ◽  
The Right ◽  
Yearly Data ◽  
Do So

The main objective of this research is to gather empirical evidence on the effects of more or less stringency and more or less risk sensitivity in regulatory capital requirements on the observed behaviour of European banks during the initial years of the financial crisis. To do so, we use the indices built in Argimón and Ruiz (2010), which capture such characteristics of capital regulation. We test their incidence using changes in yearly data for individual banks for 25 countries of the European Union covering the period 2007-2009. Our results show that more stringency and risk sensitivity in capital regulation resulted in higher capital increases, with limited effect on risk taking. However, for well capitalized banks, higher risk sensitivity resulted in higher capital and higher risk, thus requiring striking the right balance, so as to lead to increased stability.

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 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.

On the Geometry of the Automorphism Group of a Free Group

1995 ◽  
Vol 27 (6) ◽  
pp. 544-552 ◽  
Author(s):  
Martin R. Bridson ◽  
Karen Vogtmann
Keyword(s):  
Automorphism Group ◽  
Free Group

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.

Linear Representations of the Automorphism Group of a Free Group

2009 ◽  
Vol 18 (5) ◽  
pp. 1564-1608 ◽  
Author(s):  
Fritz Grunewald ◽  
Alexander Lubotzky
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Linear Representations
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