Bestvina measuring formula

In the mathematical field of geometric group theory that calculates Bestvina measuring formula (even set of Bestvina and measurement ), the dimension of the edge of a hyperbolic group from their group cohomology . It was proven by Mladen Bestvina and Geoffrey Mess .

Bestvina and Mess's theorem

Let be a hyperbolic group , then for the dimension of its edge : ${\displaystyle \Gamma }$ ${\displaystyle \partial _{\infty }\Gamma }$

${\displaystyle dim(\partial _{\infty }\Gamma )=\max \left\{n\colon H^{n}(\Gamma ,\mathbb {Z} \Gamma )\not =0\right\}.}$

In particular, applies to torsion-free hyperbolic groups

${\displaystyle dim(\partial _{\infty }\Gamma )=cd(\Gamma ),}$

where denotes the cohomological dimension of the group . ${\displaystyle cd(\Gamma )}$ ${\displaystyle \Gamma }$

Z sets

The Bestvina-Mess formula follows from the isomorphism of - modules (for any ring ) proven by Bestvina and Mess : ${\displaystyle R\Gamma }$ ${\displaystyle R}$

${\displaystyle H^{i}(\Gamma ,R\Gamma )\cong {\check {H}}(\partial \Gamma ,R),}$

where the right side denotes the Čech cohomology of the edge with coefficients in the ring . ${\displaystyle \partial \Gamma }$ ${\displaystyle R}$

This in turn follows from the following theorem proved by Bestvina and Mess in 1991.

Let be the rep complex of the hyperbolic group . Then there is an absolute retract and a quantity in . ${\displaystyle P(\Gamma )}$ ${\displaystyle \Gamma }$ ${\displaystyle {\overline {P(\Gamma )}}:=P(\Gamma )\cup \partial _{\infty }\Gamma }$ ${\displaystyle \partial _{\infty }\Gamma }$ ${\displaystyle {\mathcal {Z}}}$ ${\displaystyle {\overline {P(\Gamma )}}}$

The latter means that for every completed subset there is a homotopy with and such that ${\displaystyle A\subset \partial _{\infty }\Gamma }$ ${\displaystyle H\colon {\overline {P(\Gamma )}}\times \left[0,1\right]\to {\overline {P(\Gamma )}}}$ ${\displaystyle H_{0}=id}$ ${\displaystyle H_{t}\vert _{A}=id}$

${\displaystyle H_{t}({\overline {P(\Gamma )}}\setminus A)\subset {\overline {P(\Gamma )}}\setminus \partial _{\infty }\Gamma }$

applies to all . ${\displaystyle t>0}$

Applications

Bestvina and Mess use their formula to prove the following theorem about the local topology of the boundary:

Be a hyperbolic group. There is a ring and a ring for which is finitely generated and not zero. When connected , then it is locally connected . ${\displaystyle \Gamma }$ ${\displaystyle R}$ ${\displaystyle i>0}$ ${\displaystyle H^{i}(\Gamma ,R\Gamma )}$ ${\displaystyle \partial _{\infty }\Gamma }$

For the fundamental groups of closed , irreducible 3-manifolds , they prove that homeomorphic to the 2-sphere and the universal superposition is homeomorphic to , and homeomorphic to the closed 3-sphere . ${\displaystyle \Gamma =\pi _{1}M}$ ${\displaystyle M}$ ${\displaystyle \partial _{\infty }\Gamma }$ ${\displaystyle {\widetilde {M}}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\widetilde {M}}\cup \partial _{\infty }\Gamma }$ ${\displaystyle B^{3}}$

In higher dimensions , the analogous proposition applies that for a torsion-free, hyperbolic group , which is the fundamental group of a closed, aspherical -manifold with and , the boundary must be homeomorphic . ${\displaystyle n\geq 6}$ ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle {\widetilde {M}}\cong \mathbb {R} ^{n}}$ ${\displaystyle {\widetilde {M}}\cup \partial _{\infty }\Gamma \cong B^{n}}$ ${\displaystyle S^{n-1}}$

literature

• M. Bestvina, G. Mess: The boundary of negatively curved groups . J. Amer. Math. Soc. 4: 469-481 (1991).

Individual evidence

1. ↑ A. Bartels, W. Lück, S. Weinberger: On hyperbolic groups with spheres as boundary . J. Diff. Geom. 86, 1-16 (2010).