论文信息 - Automata and Square Complexes

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.

Yair Glasner | Shahar Mozes

[1] Groups acting on CAT(0) cube complexes , 1997, math/9702231.

[2] A. Lubotzky,et al. Superrigidity for the commensurability group of tree lattices , 1994 .

[3] Thomas Schick,et al. On a question of Atiyah , 2000 .

[4] J. Rogawski. Automorphic representations of unitary groups in three variables , 1990 .

[5] Frank Thomson Leighton,et al. Finite common coverings of graphs , 1982, J. Comb. Theory, Ser. B.

[6] Marc Burger,et al. Lattices in product of trees , 2000 .

[8] Said Sidki,et al. The Generation of GL(n, Z) by Finite State Automata , 1998, Int. J. Algebra Comput..

[9] Andrzej Żuk,et al. The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum , 2001 .

[10] M. Vignéras. Arithmétique des Algèbres de Quaternions , 1980 .

[11] Said Sidki,et al. Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity , 2000 .

[12] M. Bridson,et al. Metric Spaces of Non-Positive Curvature , 1999 .

[13] D. Mumford. An Algebraic Surface with \(K\) ample, \((K^2)= 9, p_g = q = 0\) , 1979 .

[14] M. Burger,et al. Finitely presented simple groups and products of trees , 1997 .

[15] Daniel T. Wise,et al. Non-positively curved squared complexes aperiodic tilings and non-residually finite groups , 1996 .

[16] H. Bass,et al. Uniform tree lattices , 1990 .

[17] Andrei S. Rapinchuk,et al. Algebraic groups and number theory , 1992 .