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Receding polar regions of a spherical building and the center conjecture

Bernhard Mühlherr, Richard M. Weiss (2013)

Annales de l’institut Fourier

We introduce the notion of a polar region of a spherical building and use some simple observations about polar regions to give elementary proofs of various fundamental properties of root groups. We combine some of these observations with results of Timmesfeld, Balser and Lytchak to give a new proof of the center conjecture for convex chamber subcomplexes of thick spherical buildings.

Relative property (T) and linear groups

Talia Fernós (2006)

Annales de l’institut Fourier

Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group Γ admits a special linear representation with non-amenable R -Zariski closure if and only if it acts on an Abelian group A (of...

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

Let Γ be a group and r n ( Γ ) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function 𝒵 Γ ( s ) = n = 1 r n ( Γ ) n - s . When Γ is an arithmetic group satisfying the congruence subgroup property then 𝒵 Γ ( s ) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place...

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