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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On free semigroups of affine maps on the real line
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by Alexander Kolpakov and Alexey Talambutsa

Proc. Amer. Math. Soc. 150 (2022), 2301-2307

DOI: https://doi.org/10.1090/proc/15832

Published electronically: March 16, 2022

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Abstract:

In this note we generalise some of the work of Klarner on free semigroups of affine maps acting on the real line by using a classical approach from geometric group theory (the Ping-Pong lemma). We also investigate the boundaries up to which the Klarner’s necessary condition for a semigroup to be related is applicable.

References

S. A. Adeleke, A. M. W. Glass, and L. Morley, Arithmetic permutations, J. London Math. Soc. (2) 43 (1991), no. 2, 255–268. MR 1111584, DOI 10.1112/jlms/s2-43.2.255
Julien Cassaigne, Tero Harju, and Juhani Karhumäki, On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 295–305. Dedicated to the memory of Marcel-Paul Schützenberger. MR 1723469, DOI 10.1142/S0218196799000199
Stephen D. Cohen, The group of translations and positive rational powers is free, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 181, 21–93. MR 1326133, DOI 10.1093/qmath/46.1.21
Stephen D. Cohen and A. M. W. Glass, Free groups from fields, J. London Math. Soc. (2) 55 (1997), no. 2, 309–319. MR 1438634, DOI 10.1112/S002461079600436X
B. Deroin, A. Navas, and C. Rivas, Groups, Orders, and Dynamics, arXiv:1408.5805, 215 pp. 2016.
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
Richard K. Guy, Unsolved Problems: Don’t Try to Solve These Problems, Amer. Math. Monthly 90 (1983), no. 1, 35–38+39–41. MR 1540108, DOI 10.2307/2975688
David A. Klarner, A sufficient condition for certain semigroups to be free, J. Algebra 74 (1982), no. 1, 140–148. MR 644223, DOI 10.1016/0021-8693(82)90010-2
D. A. Klarner, Sets of integers closed under unary, affine operations, Mathematical Sciences Department, SUNY Binghamton, Preprint, 1978.
David A. Klarner, Jean-Camille Birget, and Wade Satterfield, On the undecidability of the freeness of integer matrix semigroups, Internat. J. Algebra Comput. 1 (1991), no. 2, 223–226. MR 1128014, DOI 10.1142/S0218196791000146
Jeffrey C. Lagarias, Erdős, Klarner, and the $3x+1$ problem, Amer. Math. Monthly 123 (2016), no. 8, 753–776. MR 3552747, DOI 10.4169/amer.math.monthly.123.8.753
MichałMisiurewicz and Ana Rodrigues, Real $3x+1$, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1109–1118. MR 2117212, DOI 10.1090/S0002-9939-04-07696-8
Samuel White, The group generated by $x\mapsto x+1$ and $x\mapsto x^p$ is free, J. Algebra 118 (1988), no. 2, 408–422. MR 969681, DOI 10.1016/0021-8693(88)90030-0