scholarly journals Finiteness properties of cubulated groups

2014 ◽  
Vol 150 (3) ◽  
pp. 453-506 ◽  
Author(s):  
G. C. Hruska ◽  
Daniel T. Wise
Keyword(s):  
Hyperbolic Groups ◽  
Cube Complex ◽  
Relatively Hyperbolic Groups ◽  
Finiteness Properties ◽  
Cube Complexes ◽  
Relatively Hyperbolic

AbstractWe give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

Finiteness properties of groups

1948 ◽  
Vol 15 (4) ◽  
pp. 1021-1032 ◽  
Author(s):  
Reinhold Baer
Keyword(s):  
Finiteness Properties

scholarly journals Homological finiteness properties of fibre products

2018 ◽  
Vol 69 (3) ◽  
pp. 835-854 ◽  
Author(s):  
Dessislava H Kochloukova ◽  
Francismar Ferreira Lima
Keyword(s):  
Fibre Products ◽  
Finiteness Properties

scholarly journals Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

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 $\mathbf{SL}_{n}(\mathbb{Z})$.

scholarly journals Finiteness Properties of Affine Difference Algebraic Groups

2020 ◽  
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.

scholarly journals Topological generation results for free unitary and orthogonal groups

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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