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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.
A NUMERICAL LOWER BOUND FOR THE SPECTRAL RADIUS OF RANDOM WALKS ON SURFACE GROUPS
, 2013
"... Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound for below for this spectral radius in Cayley graphs with finitely many cone types (incl ..."
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Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound for below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.
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.