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Elementary Equivalence of Right-angled Coxeter Groups and Graph Products of Finite Abelian Groups
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kazachkov, Ilya V. |
| Copyright Year | 2008 |
| Abstract | We show that graph products of finite abelian groups are elementarily equivalent if and only if they are ∃∀-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic. The notion of elementary equivalence is fundamental in model theory. One of the most natural problems about elementary equivalence, is given a class of algebraic systems, to understand which systems in this class are elementarily equivalent, i.e. to classify algebraic systems up to elementary properties. This problem in the case of groups is usually rather hard and there are only few examples known when this problem has a satisfactory solution. For the class of abelian groups, this is a well-known result of W. Szmielew (1955). For the class of ordered abelian groups this problem was studied by A. Robinson and E. Zakon (1960), M. Kargapolov (1963) and Yu. Gurevich (1964). A. Malcev (1961) solved this problem for classical linear groups. His approach was generalised by E. Bunina and A. Mikahlëv to other linear, algebraic and Chevalley groups. The problem of classifying linear groups over integers up to elementary properties was studied by V. Durnev (1995). For certain nilpotent groups this problem was studied by O. Belegradek, R. Deborah, A. Miasnikov, F. Oger, V. Remelsennikov. For certain free operator groups, this problem was solved by A. Miasnikov and V. Remeslennikov (1987). Around 1945 it was conjectured by A. Tarski, that elementary theories of free non-abelian groups of different rank coincide. This conjecture is now known as Tarski’s problem and has recently been solved by O. Kharlampovich and A. Miasnikov (2006), and, independently, by Z. Sela (2006). The classification of torsionfree hyperbolic groups up to elementary properties has been recently announced by Z. Sela. In this paper we address the classification of graph products of finite abelian groups up to elementary properties. We prove that two graph products of finite abelian groups are elementarily equivalent if and only if they are ∃∀-equivalent if 2000 Mathematics Subject Classification. Primary 20E06, 20F55, 03C07; Secondary 20E99. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0810.4870v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Game theory Graph - visual representation Graph product Linear algebra Mathematics Subject Classification Neoplasm Metastasis Order (action) Turing completeness |
| Content Type | Text |
| Resource Type | Article |
