Mathematics > Group Theory
[Submitted on 11 Mar 2014 (v1), last revised 1 Feb 2016 (this version, v2)]
Title:On the penetration distance in Garside monoids
View PDFAbstract:We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irreducible Artin monoids of spherical type, extending a result by Caruso.
Our results establish that the expected value of the penetration distance $pd(x,y)$ in an irreducible Artin monoid of spherical type is bounded independently of the length of $x$, if $x$ is chosen uniformly among all elements of given canonical length and $y$ is chosen uniformly among all atoms; the latter in particular explains observations made by Thurston in the context of the braid group, and it shows that all irreducible Artin monoids of spherical type exhibit an analogous behaviour. Our results also give an affirmative answer to a question posed by Dehornoy.
Submission history
From: Volker Gebhardt [view email][v1] Tue, 11 Mar 2014 18:05:26 UTC (28 KB)
[v2] Mon, 1 Feb 2016 21:33:45 UTC (35 KB)
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