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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimum simplicial complexes with given abelian automorphism group

Author: Zevi Miller
Journal: Trans. Amer. Math. Soc. 271 (1982), 689-718
MSC: Primary 05C65; Secondary 20B25
MathSciNet review: 654857
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Abstract: Let $ K$ be a pure $ n$-dimensional simplicial complex. Let $ {\Gamma _0}(K)$ be the automorphism group of $ K$, and let $ {\Gamma _n}(K)$ be the group of permutations on $ n$-cells of $ K$ induced by the elements of $ {\Gamma _0}(K)$. Given an abelian group $ A$ we consider the problem of finding the minimum number of points $ M_0^{(n)}(A)$ in $ K$ such that $ {\Gamma _0}(K) \cong A$, and the minimum number of $ n$-cells $ M_1^{(n)}(A)$ in $ K$ such that $ {\Gamma _n}(K) \cong A$. Write $ A = {\prod _{{p^\alpha }}}{\mathbf{Z}}_{{p^\alpha }}^{e({p^\alpha })}$, where each factor $ {{\mathbf{Z}}_{{p^\alpha }}}$ appears $ e({p^\alpha })$ times in the canonical factorization of $ A$. For $ A$ containing no factors $ {{\mathbf{Z}}_{{p^\alpha }}}$ satisfying $ {p^\alpha } < 17$ we find that $ M_1^{(n)}(A) = M_0^{(2)}(A) = {\sum _{{p^\alpha }}}{p^\alpha }e({p^\alpha })$ when $ n \geqslant 4$, and we derive upper bounds for $ M_1^{(n)}(A)$ and $ M_0^{(n)}(A)$ in the remaining possibilities for $ A$ and $ n$.

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Article copyright: © Copyright 1982 American Mathematical Society