The ω\omega-Borel invariant for representations into SL(n,Cω)(n,\mathbb{C}_\omega)

  • Alessio Savini

    Università di Bologna, Italy
The $\omega$-Borel invariant for representations into SL$(n,\mathbb{C}_\omega)$ cover
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Let Γ\Gamma be the fundamental group of a complete hyperbolic 33-manifold MM with toric cusps. By following [3] we define the ω\omega-Borel invariant βnω(ρω)\beta_n^\omega(\rho_\omega) associated to a representation ρω ⁣:ΓSL(n,\Cω)\rho_\omega\colon \Gamma \rightarrow \mathrm{SL}(n,\C_\omega), where \Cω\C_\omega is a field introduced by [18] which can be constructed as a quotient of a suitable subset of \CN\C^\N with the data of a non-principal ultrafilter ω\omega on N\N and a real divergent sequence λl\lambda_l such that λl1\lambda_l \geq 1.

Since a sequence of ω\omega-bounded representations ρl\rho_l into SL(n,\C)\mathrm{SL}(n,\C) determines a representation ρω\rho_\omega into SL(n,\Cω)\operatorname{SL}(n,\C_\omega), for n=2n=2 we study the relation between the invariant β2ω(ρω)\beta^\omega_2(\rho_\omega) and the sequence of Borel invariants β2(ρl)\beta_2(\rho_l). We conclude by showing that if a sequence of representations ρl ⁣:ΓSL(2,\C)\rho_l\colon \Gamma \rightarrow \mathrm{SL}(2,\C) induces a representation ρω ⁣:ΓSL(2,\Cω)\rho_\omega\colon \Gamma \rightarrow \operatorname{SL}(2,\C_\omega) which determines a reducible action on the asymptotic cone Cω(H3,d/λl,O)C_\omega(\mathbb{H}^3,d/\lambda_l,O) with non-trivial length function, then it holds β2ω(ρω)=0\beta^\omega_2(\rho_\omega)=0.

Cite this article

Alessio Savini, The ω\omega-Borel invariant for representations into SL(n,Cω)(n,\mathbb{C}_\omega). Groups Geom. Dyn. 13 (2019), no. 3, pp. 981–1006

DOI 10.4171/GGD/511