Minimum simplicial complexes with given abelian automorphism group

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 $n$.