Stability in the homology of congruence subgroups

By Andrew Putman. 2012

71 citations recorded by the source corpus.

The homology groups of many natural sequences of groups $$\{G_n\}_{n=1}^{\infty }$${Gn}n=1∞ (e.g. general linear groups, mapping class groups, etc.) stabilize as $$n \rightarrow \infty $$n→∞. 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 representation stability. We prove that the homology groups of congruence subgroups of $${{\mathrm{GL}}}_n(R)$$GLn(R) (for almost any reasonable ring $$R$$R) satisfy a strong version of representation stability that we call 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 representation stability. Our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f9e47d02f05ce542af90a9e04c9f21910ed0aec

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Citations in the corpus

Representation stability and finite linear groups (2014, background): “ymptotic structure theorem does not provide bounds for when it begins. This is an unavoidable artifact of our proof, which make essential use of local Noetherianity. The results of the ﬁrst author in [Pu2] concerning central stability for the homology groups of congruence subgroups (as representatio”
Higher-order representation stability and ordered configuration spaces of manifolds (2016, methodology): “des a very general context for formulating representation stability for sequences of symmetric group representations. We then discuss the relationship between Putman’s central stability chain complex [Put15] and Farmer’s complex of injective words [Far79] in Section2.2. In Section2.3, we give a new ”
FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “Putman [Put12] independently developed a theory that extends representation stability to positive characteristic.”
Homology of FI-modules (2015, methodology): “M / I Z / , but its formulation here is a combination of [12, Theorem B] and our earlier theorem with Farb and Nagpal [5, Theorem 1.6].”
FI-modules and the cohomology of modular representations of symmetric groups (2015, methodology): “ule V=(Vn)n>0: there exists some N> 0 such that if n> Nthen Vncan be described in terms of Vj, j6 N. Such an inductive description had appeared before in characteristic 0 (see Putman’s paper [Pu]) 2 and has several applications. In this paper, the techniques introduced in [CEFN] are used ex”