Mathematics – Algebraic Topology
Scientific paper
2012-01-23
Mathematics
Algebraic Topology
36 pages, 1 figure; a few small corrections
Scientific paper
The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g.\ general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena {\em representation stability}. We prove that the homology groups of congruence subgroups of $\GL_n(R)$ (for almost any reasonable ring $R$) and mapping class groups of manifolds with marked points satisfy a strong version of representation stability that we call {\em central stability}. The definition of central stability is very different from Church-Farb's definition of representation stability (it is defined via a universal property), but we prove that it implies Church-Farb's definition of representation stability. Our main tool is a new machine analogous to the classical homological stability machine for proving central stability.
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