Bass – Tits minimization of automata , quotients of trees and diameters 3 Lisa Carbone

9 Let X be a tree and letG=Aut(X), Bass and Tits have given an algorithm to construct the ‘ultimate quotient’ of X by G starting with any quotient of X, an ‘edge-indexed’ graph. Using a sequence of 11 integers that we compute at consecutive steps of the Bass–Tits (BT) algorithm, we give a lower bound on the diameter of the ultimate quotient of a tree by its automorphism group. For a tree X with finite 13 quotient, this gives a lower bound on the minimum number of generators of a uniform X-lattice whose quotient graph coincides withG\X. This also gives a criterion to determine if the ultimate quotient of 15 a tree is infinite. We construct an edge-indexed graph (A, i) for a deterministic finite state automaton and show that the BT algorithm for computing the ultimate quotient of (A, i) coincides with state 17 minimizing algorithm for finite state automata. We obtain a lower bound on the minimum number of states of the minimized automaton. This gives a new proof that language for the word problem in a 19 finitely generated group is regular if and only if the group is finite, and a new proof that the language of the membership problem for a subgroup is regular if and only if the subgroup has finite index. 21 © 2005 Published by Elsevier B.V. MSC: primary 20F32; secondary 22F50 23 ∗ Corresponding author. E-mail address: [email protected] (L. Carbone). 1 The first author was supported in part by NSF Grant #DMS-0401107. 2 The second author was supported in part by a research award for undergraduates at Harvard, and an NSF Graduate Research Fellowship at the University of Michigan. 0022-4049/$ see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.jpaa.2005.04.013 UN CO RR EC TE D PR OO F 2 L. Carbone, D. Clark / Journal of Pure and Applied Algebra ( ) – JPAA3227 ARTICLE IN PRESS