On the penetration distance in Garside monoids
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- 作者：Volker Gebhardt ; v.gebhardt@westernsydney.edu.au" class="auth_mail" title="E-mail the corresponding author ; Stephen Tawn stephen@tawn.co.uk" class="auth_mail" title="E-mail the corresponding author
- 关键词：primary ; 20F36 ; 43A07 ; secondary ; 20F69 ; 60B15
- 刊名：Journal of Algebra
- 出版年：2016
- 出版时间：1 April 2016
- 年：2016
- 卷：451
- 期：Complete
- 页码：544-576
- 全文大小：709 K
文摘
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.