I am studying Bestvina and Mess's results on the boundary of hyperbolic groups [The Boundary of Negatively Curved Groups. Journal of the American Mathematical Society Vol. 4, No. 3 (Jul., 1991), pp. 469-481], but I am having trouble with the proof of Corollary 1.4, especially the final part (starting from "Assuming $dim_R\partial\Gamma=i\geq k$...").
Could anybody spell the proof out please? Or even point out a more detailed version of the proof?
Question edited according to the comment below.
In particular, these are my doubts:
- The statement of Corollary 1.4 says "(a) $dim_R\partial\Gamma = \max\{n\mid H^n (\Gamma; R\Gamma) \neq0 \}$". I think, in view of Corollary 1.3, it should be "(a) $dim_R\partial\Gamma = \max\{n\mid H^n (\Gamma; R\Gamma) \neq0 \}-1$", which seems also coherent with Corollary 1.4(c).
- In the 2nd part of the proof of Corollary 1.4(a), i.e. after the proof of the claim, they write "...and construct closed sets $\partial\Gamma = X \supset B_0 \supset B_1 \supset B_2$ as in the proof of Proposition 2.6". Do they mean $P(\Gamma)\cup\partial\Gamma = X \supset B_0 \supset B_1 \supset B_2$? (Setting $\partial\Gamma=X$ seems incoherent with the following formulas)
- If this is the case, how do they construct the $B_i$'s? Are $B_2=h_1(X)$, $B_1=h(B_2\times[0,1])$, $B_0=h(B_1\times[0,1])$, where $h:X\times[0,1]$ is a homotopy satisfying $h_0=id$, $h_t|_A=inclusion$, $h_t(X \setminus A) \subset (X \setminus \partial\Gamma)$ for $t>0$?
- It seems to me that the general scheme of the proof can be summarized in the following diagram (all cohomology groups with coefficients in $R$):
$\require{AMScd}$
\begin{CD}
H^i(X,B_0) @>a_0>> H^i(\partial\Gamma\cup_AB_0,B_0)@>j_0>> H^{i+1}(X,\partial\Gamma\cup_AB_0) @>\cong>> H_c^{i+1}(P(\Gamma),B_0\cap P(\Gamma))\\
@V b_0 V V @Vc_0V V @Vd_0VV @Ve_0VV\\
H^i(X,B_1) @>a_1>> H^i(\partial\Gamma\cup_AB_1,B_1)@>j_1>> H^{i+1}(X,\partial\Gamma\cup_AB_1) @>\cong>> H_c^{i+1}(P(\Gamma),B_1\cap P(\Gamma))\\
@V b_1 V V @Vc_1V V @. @.\\
H^i(X,B_2) @>a_2>> H^i(\partial\Gamma\cup_AB_2,B_2)@>j_2>> H^{i+1}(X,\partial\Gamma\cup_AB_2) @>\cong>> H_c^{i+1}(P(\Gamma),B_2\cap P(\Gamma))\\
\end{CD}
- The (first 3 terms of the horizontal rows are the cohomology long exact sequences of the triples $(X, \partial\Gamma\cup_AB_i,B_i)$.
- The isomorphisms between groups in the last 2 columns seem to me an application of excision for $H^*_c$, taking into account that $X$ is compact, so $H^*_c(X,...)=H^*(X,...)$: is this correct?
- The vertical maps are induced by the inclusions $B_2\subset B_1\subset B_0$, so they map a cohomology class $[\varphi]$ to $[\varphi]$ (but of course the classes may be different, even if the representative can be chosen to be the same). This is why the diagram commutes.
- $c_0$ and $c_1$ are isomorphisms (by excision, each of the groups in the 2nd column is isomorphic to $H^i(\partial\Gamma, A)$).
- If $b_0=0$ then $j_0$ is injective: this is just diagram chasing. But why is $b_0=0$?
- Analogously, if $b_1=0$ then $j_1$ is injective.
- Hence, if $b_0=b_1=0$, each of $H^{i+1}(X,\partial\Gamma\cup_AB_0)$ and $H^{i+1}(X,\partial\Gamma\cup_AB_1)$ contains a copy of $H^i(\partial\Gamma, A)$; by commutativity of the diagram, $d_0$ maps one copy isomorphically onto the other, so $d_0\neq 0$.
- On the other side, by the preceding claim (first part of the proof of Corollary 1.4), $e_0=0$, which contradicts $d_0\neq 0$. Is this summarization correct?
- Also, it seems to me that the only role played by the group $\Gamma$ is to ensure that the claim in the previous part of the proof holds, namely that for any $i>k$ there exists an integer $K$ such that every $i$-cocycle $z$ is the coboundary of a cochain whose support is contained in the combinatorial $K$-neighbourhood of the support of $z$. This is needed to ensure that the map $e_0$ is zero. Does indeed the rest of the proof only depend on the fact that the $1$-skeleton of $P(\Gamma)$ is a hyperbolic space and $\partial\Gamma$ is its boundary in purely topological terms?
Thank you so much!
