scholarly journals An isoperimetric function for Bestvina-Brady groups

2008 ◽  
Vol 40 (3) ◽  
pp. 384-394 ◽  
Author(s):  
Will Dison
Keyword(s):  
Isoperimetric Function

scholarly journals ISOPERIMETRIC FUNCTIONS OF GROUPS ACTING ON Lδ-SPACES

2007 ◽  
Vol 49 (1) ◽  
pp. 23-28
Author(s):  
JON CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Metric Space ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Generated Group ◽  
Isoperimetric Functions ◽  
Finitely Presented

Abstract.A finitely generated group acting properly, cocompactly, and by isometries on an Lδ-metric space is finitely presented and has a sub-cubic isoperimetric function.

scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

scholarly journals Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

2021 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli F. Tedeev
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Degenerate Parabolic Equations ◽  
Isoperimetric Function ◽  
Finite Speed Of Propagation ◽  
The Cauchy Problem ◽  
Inhomogeneous Density ◽  
Degenerate Parabolic

AbstractWe consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

1998 ◽  
Vol 08 (02) ◽  
pp. 235-294 ◽  
Author(s):  
Jean-Camille Birget
Keyword(s):  
Word Problem ◽  
Time Complexity ◽  
Linear Time ◽  
Algorithm Design ◽  
Embedding Theorem ◽  
Algebraic Characterization ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Presented

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem: Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) ≥T(n)+T(m)). Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is O (T(∣x∣+∣y∣)2). Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(To)) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(To)). In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function ≤ D (where D(n) ≥ n), then the word problem of S has nondeterministic time complexity O(D). The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T) O(1) )) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) O(1) ) isoperimetric function. An algorithmic problem L is in NP (or more generally, in NTIME((T) O(1) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) O(1) ) isoperimetric function. In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

scholarly journals Residually free groups do not admit a uniform polynomial isoperimetric function

2020 ◽  
Vol 148 (10) ◽  
pp. 4203-4212
Author(s):  
Claudio Llosa Isenrich ◽  
Romain Tessera
Keyword(s):  
Free Groups ◽  
Isoperimetric Function
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