Finiteness Properties of Some Families of GP-Trees

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

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Finiteness Properties of Affine Difference Algebraic Groups

International Mathematics Research Notices ◽

10.1093/imrn/rnaa177 ◽

2020 ◽

Cited By ~ 1

Author(s):

Michael Wibmer

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Difference Equations ◽

General Linear Group ◽

Algebraic Groups ◽

The Matrix ◽

Algebraic Difference ◽

The Difference ◽

Linear Differential ◽

Finiteness Properties

Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

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Topological generation results for free unitary and orthogonal groups

International Journal of Mathematics ◽

10.1142/s0129167x20500032 ◽

2019 ◽

Vol 31 (01) ◽

pp. 2050003

Author(s):

Alexandru Chirvasitu

Keyword(s):

Unitary Group ◽

Free Groups ◽

Orthogonal Groups ◽

Factorization Property ◽

Residually Finite ◽

Inductive Proof ◽

Classical Counterpart ◽

Lower Rank ◽

Maximal Tori ◽

Finiteness Properties

We show that for every [Formula: see text] the free unitary group [Formula: see text] is topologically generated by its classical counterpart [Formula: see text] and the lower-rank [Formula: see text]. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all [Formula: see text], also hold at [Formula: see text]. Specifically, all discrete quantum duals [Formula: see text] and [Formula: see text] are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, [Formula: see text] are topologically generated by [Formula: see text] and their maximal tori [Formula: see text] (dual to the free groups on [Formula: see text] generators) and similarly, [Formula: see text] are topologically generated by [Formula: see text] and their tori [Formula: see text].

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