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Multiplicative measures on free groups
 INT. J. ALGEBRA COMP
, 2002
"... 1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discret ..."
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Cited by 14 (3 self)
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1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discrete infinite groups, as well as in the analysis of the behaviour of heuristic (e.g. genetic) algorithms for infinite groups [22, 23]. In most computerbased computations in finitely presented groups G = F/R the elements are represented as freely reduced words in the free group F, with procedures for comparing their images in the factor group G = F/R. Therefore the ambient algebraic structure in all our considerations is the free group F = F(X) on a finite set X = {x1,...,xm}. We identify F with the set of all freely reduced words in the alphabet X ∪X −1, with the multiplication given by concatenation of words with the subsequent free reduction. The most natural and convenient way to generate pseudorandom elements
On the cogrowth of Thompson’s group
 F . Groups Complex. Cryptol
"... We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. 1 ..."
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We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. 1
Quotient tests and random walks in computational group theory
, 2005
"... For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few sim ..."
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Cited by 3 (1 self)
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For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few simple case studies of quotient tests for the classical identity, word, conjugacy problems in groups. We attempt to combine a rigorous analytic study with the assessment of algorithms from the practical point of view. It appears that, in case of finite quotient groups G/K, the efficiency of the quotient test very much depends on the mixing times for random walks on the Cayley graph of G/K.
RANDOM SAMPLING OF TRIVIALS WORDS IN FINITELY PRESENTED GROUPS
"... Abstract. We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution pi(w) = (w+ 1)αβw  · Z−1 where w  ..."
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Abstract. We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution pi(w) = (w+ 1)αβw  · Z−1 where w  is the length of a word w, α and β are parameters of the algorithm, and Z is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which then allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the BaumslagSolitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), and Richard Thompson’s group F. 1.
ON TRIVIAL WORDS IN FINITELY PRESENTED GROUPS
"... all great men of generating functions. Abstract. We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of s ..."
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all great men of generating functions. Abstract. We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for BaumslagSolitar groups BS(N,N) = 〈a, baN b = baN 〉 and prove that their cogrowth rates are algebraic numbers. 1.
On the cogrowth of Thompson’s group F 1
"... We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. ..."
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We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F.