Amenability of groups is characterized by Myhill’s theorem. (English) Zbl 1458.37017
The paper deals with a converse to J. Myhill’s “Garden-of-Eden” theorem [Proc. Am. Math. Soc. 14, 685–686 (1963; Zbl 0126.32501)]. Let \(G\) be a non-amenable group. The author proves that there exists a cellular automaton carried by \(G\) that admits gardens of Eden but no mutually erasable patterns. The author then proves the following theorem. Let \(G\) be a group and let \(\mathbb{K}\) be a field such that \(\mathbb{K}G\) has no zero divisors. Then \(G\) is amenable if and only if \(\mathbb{K}G\) is an Ore domain.
Reviewer: Hasan Akin (Gaziantep)
MathOverflow Questions:
What is the current status of the Kaplansky zero-divisor conjecture for group rings?MSC:
| 37B15 | Dynamical aspects of cellular automata |
| 37B10 | Symbolic dynamics |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 16U20 | Ore rings, multiplicative sets, Ore localization |
Citations:
Zbl 0126.32501References:
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