We describe the structure of the lattice of normal subgroups of the group of local isometries of the boundary of a spherically homogeneous tree LIsom ∂T. It is proved that every normal subgroup of this group contains its commutant. We characterize the quotient group of the group LIsom ∂T by its commutant.
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References
V. Nekrashevych, Self-Similar Groups, American Mathematical Society, Providence, RI (2005).
V. I. Sushchanskii, Normal Structure of Isometry Groups of Semifinite Baire Metrics. Infinite Groups and Related Algebraic Structures [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1993).
H. Bass, M. V. Otero-Espinar, D. Rockmore, and C. Tresser, Cyclic Renormalization and Automorphism Groups of Rooted Trees, Springer, Berlin (1995).
Y. Muntyan and V. I. Sushchansky, Normal Structure of the Big Group of a Spherical Homogeneous Rooted Tree, Preprint, TAMU (2006).
Ya. V. Lavrenyuk and V. I. Sushchanskii, “Normal structure of a group of local isometries of the boundary of a spherically homogeneous tree,” Dopov. Nats. Akad. Nauk Ukr., No. 3, 20–24 (2007).
O. E. Bezushchak and V. I. Sushchanskii, “l-Wreath products and isometries of generalized Baire metrics,” Ukr. Mat. Zh., 43, No. 7/8, 1031–1038 (1991).
R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, “Automata, dynamical systems, and groups,” Tr. Mat. Inst. Ros. Akad. Nauk, 231, 134–214 (2000).
Ya. Lavrenyuk, “On automorphisms of local isometry groups of compact ultrametric spaces,” Int. J. Algebra Comput., 15, No. 5–6, 1013–1024 (2005).
Ya. Lavrenyuk, “Classification of the local isometry groups of rooted tree boundaries,” Algebra Discrete Math., No. 1 (2007).
Ya. V. Lavrenyuk and V. I. Sushchansky, “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees,” Alg. Discrete Math., No. 4, 33–49 (2003).
N. V. Kroshko and V. I. Sushchansky, “Direct limits of symmetric and altering groups with strictly diagonal embeddings,” Arch. Math., 71, 173–182 (1998).
A. G. Kurosh, The Theory of Groups [in Russian], Nauka, Moscow (1967).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1350–1356, October, 2008.
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Lavrenyuk, Y.V., Sushchanskii, V.I. Lattice of normal subgroups of a group of local isometries of the boundary of a spherically homogeneous tree. Ukr Math J 60, 1574–1580 (2008). https://doi.org/10.1007/s11253-009-0154-8
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DOI: https://doi.org/10.1007/s11253-009-0154-8