Free products of cyclic groups in groups of infinite unitriangular matrices
Abstract
Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.
For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.
It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.
A few open questions concerning properties of presented examples and other types of faithful representations are formulated.
References
W. Holubowski, Free subgroups of the group of infinite unitriangular matrices, Internat. J. Algebra Comput., 13 (2003), no.1, 81–86. https://doi.org/10.1142/S0218196703001341
Ya. Lavrenyuk, V. Mazorchuk, A. Oliynyk, V. Sushchansky, Faithful group actions on rooted trees induced by actions of quotients, Comm. Algebra, 35 (2007), no.11, 3759–3775. https://doi.org/10.1080/00914030701410237
N. Mazza, The pro-p group of upper unitriangular matrices, J. Pure Appl. Algebra, 221 (2017), no.12, 2928–2952. https://doi.org/10.1016/j.jpaa.2017.02.009
A. Oliynyk, V. Sushchansky, A free group of infinite unitriangular matrices, Math. Notes, 67 (2000), no.3, 320–324. https://doi.org/10.1007/BF02676668
A. Oliynyk, V. Sushchansky, Representations of free products by infinite unitriangular matrices over finite fields, Internat. J. Algebra Comput., 14 (2004), no.5–6, 741–749. https://doi.org/10.1142/ S0218196704001931
A. Oliynyk, Free products of cyclic 2-groups in groups of infinite unitriangular matrices, Mat. Visn. Nauk. Tov. Im. Shevchenka, 9 (2012), 231–237. (in Ukrainian) http://journals.iapmm.lviv.ua/ojs/index.php/MBSSS/article/view/220
A. Oliynyk, V. Prokhorchuk, Amalgamated free product in terms of automata constructions. Comm. Algebra, 50 (2022), no.2, 740–750. https://doi.org/10.1080/00927872.2021.1967965
D. Savchuk, S. Sidki, Affine automorphisms of rooted trees. Geom. Dedicata, 183 (2016), 195–213. https://doi.org/10.1007/s10711-016-0154-4
R. S lowik, Free and non-free subgroups of the group ${rm UT}(infty, mathbb{Z})$. Comm. Algebra, 41 (2013), no.4, 1350–1364. https://doi.org/10.1080/00927872.2011.633139
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