# Aurel: Hyperbolic Scissors Congruence and the Bloch-Wigner Sequence (Monday, June 13) :::{.remark} References: [@DS82, p. 82]. Outline for the talk: - Hyperbolic space and the half-plane model. - The extended hyperbolic place, $P(\bar \HH)$, and ? - ? ::: ## Hyperbolic Space :::{.definition title="The half-plane model"} $\HH^n \da \ts{\tv{x_1,\cdots, x_n} \in \RR_n \st x_n > 0}$, which just excludes one axis. Geodesics are lines or circles with endpoints on $\RR^{n-1}$, and $\bd \HH^n = \RR^n\union\ts{\infty}$. We define $\bar \HH^n \da \HH^n \union \bd \HH^n$. ::: :::{.definition title="Hyperbolic scissors congruence groups"} $P(\HH^n)$ is the free abelian group on symbols $[P]$ where $P\injects \HH^n$ is a polytope with relations - $[P] - [P'] - [P'']$ when $P' \uplus P'' = P$. - $[P] = [gP]$ for all $g\in G$. ::: :::{.theorem title="Zylev"} If $[P] = [Q]$ in $P(\HH^n)$ then $P, Q$ are scissors congruent. ::: :::{.example title="?"} Note that this does not hold for $\bar \HH^n$. Consider the following triangle: ![](figures/2022-06-13_16-42-05.png) Note that $2[\Delta] = [\Delta] + [P]$ but $\Delta$ is not SC to $P$ since $\Delta$ has an ideal vertex but $P$ does not. ::: :::{.theorem title="DS82, Theorem 2"} The inclusion $\HH^n \injects \bar\HH^n$ induces an isomorphism $P(\HH^n) \iso P(\bar\HH^n)$. ::: :::{.proof title="Sketch"} Write $P(X) \cong H_0(G; \St(X)^t)$ where $\St(X) = \tilde H_{n-1}(\tau(X); \ZZ)$. For any ideal point $p\in \bd\HH^n$, let $\tau(\bar \HH^n, p)$ be the Tits complex of flags containing $p$, and note that $\St(\bar \HH^n) = \tilde H_{n-2}(?)$. :::{.lemma title="?"} There is a SES \[ \St(\HH^n) \injects \St(\bar \HH^n) \surjects \coprod_{p\in \bd \HH^n} \St(\bar\HH^n, p) .\] The proof follows from considering the LES for the pair $(T(\HH\bar ), T(\HH))$. ::: Taking the LES in homology, it suffices to show \[ H_k\qty{ G, \coprod_{p\in \bd\HH^n} \St(\bar\HH^n, p)^t } = 0, \qquad k=0,1 .\] It turns out that this is zero for all $k$. By Shapiro's lemma, one can write \[ H_k\qty{ G, \coprod_{p\in \bd\HH^n} \St(\bar\HH^n, p)^t} \iso H_*(\Sim(n-1), \St(\RR^{n-1})) .\] One can conclude using the Hochschild-Serre spectral sequence for \[ 1 \to T(n-1) \to \Sim(n-1) \to \Sim_0(n-1) \to 1 ,\] where $\Sim_0$ are similarities fixing the origin. ::: :::{.remark} Coping homological definitions, we can define $G \da \ro{\Isom(\HH_n)}{\bd \HH_n}$ and \[ P(\bd \HH^n) \da H_0(G; \St(\bd \HH^n)^t), \qquad \complex{C}(\bd \HH^n)/ \complex{C}(\bd \HH^n)^{n-1} .\] Explicitly, this is free on $(n+1)\dash$tuples of ideal points $\vector a \da (a_0, \cdots, a_n)$ with the following relations: - $\vector a = 0$ iff it lies in a hyperplane - $\sum (-1)^i (a_0, \cdots, \hat a_i, \cdots a_n) = 0$. - $(ga_0, \cdots, ga_n) = \det(g)\vector a$. ::: :::{.proposition title="DS82 3.7"} There is an exact sequence \[ H_1(\Orth_n, \St(S^{n-1})^t ) \to P(\bd \HH^n) \to P(\bd \HH^n) \surjects H_0(\Orth_n; \St(S^{n-1})^t ) .\] ::: :::{.proof title="?"} Similar to the previous theorem, but use a different SES: \[ \St(\bd \HH^n) \injects \St(\bar \HH^n) \surjects \coprod_{p\in \HH^n} \St(\HH^n, p) ,\] noting that the last term is now over only interior points. ::: :::{.example title="?"} It turns out that $P(\HH^2) \iso P(\bar\HH^2)$. In this case, $\bd \HH^2 = \RR\union\ts{\infty}$, $G = \PSL_2(\RR) \semidirect C_2$, and $G\actson \HH^2$ by \[ \matt a b c d x = {ax+b\over cx+d} \] and $C_2$ acts by $-1$. In this case, $P(\bd \HH^2) \cong \ZZ$, and we get a SES \[ \ZZ\injects P(\bar\HH^2) \surjects \RR/\pi\ZZ \] and $P(\bar \HH^2) \cong P(\HH^2) \cong \RR$. ::: # Bloch-Wigner and $P(\HH^3)$ :::{.remark} Using the upper half-space model for $\HH^3$ yields $G \cong \PSL_2(\CC)\semidirect C_2$, identifying $\bd \HH^3 \cong \CP^1$. Then $C_2$ acts by $z\mapsto -\zbar$ and $\PSL_2(\CC)$ acts by \[ \matt a b c d z = {az+b \over cz+d} .\] Let $\PP_\CC$ denote the coinvariants of the action on $\bd \HH^3$. We get a new relation: - $\vector a = 0$ iff it lies in a hyperplane - $\sum (-1)^i (a_0, \cdots, \hat a_i, \cdots a_n) = 0$. - $(ga_0, \cdots, ga_n) = \vector a$. For any $\vector a\in \HH^3$, there is a $g\in G$ such that \[ \vector a = g(\infty, 0, 1, z), \qquad z \da {a_0 - a_2 \over a_0-a_3}\cdot {a_1-a_3\over a_1-a_2}\in \CC\smts{0, 1} .\] ::: :::{.theorem title="Bloch-Wigner"} There is an exact sequence \[ \QQ/\ZZ \injectsvia{\alpha} H_3(\SL_2(\CC)) \mapsvia{F} P_\CC \mapsvia{\gamma} \Extprod^2 \CC\units \surjects \K_2(\CC)\iso H_2(\SL_2(\CC); \ZZ) .\] ::: :::{.remark} Note that: - Elements in $\K_2(\CC)$ are written in brackets $\ts{a, b}$ due to the isomorphism $\K_M(\CC) \iso \K_2(\CC)$. - $\delta (a\wedge b) = \ts{a, b}$. - $\alpha$ is induced by \[ \QQ/\ZZ &\to \SL_2(\CC) \\ z &\mapsto \matt z 0 0 {z\inv} .\] - $\beta[g_1 |g_2|g_3] = (\infty, g_1 \infty, g_1g_2\infty, g_1g_2g_3\infty )$ - $\gamma \ts{z} = z\wedge (1-z)$. - Proving this involves a hypercohomology spectral sequence, and $\gamma$ is an awful transgression map. ::: :::{.remark} We can define $P_F$ in a similar way for any field $F$. ::: :::{.theorem title="DS82 5.1"} When $F$ is algebraically closed, $P_F$ is a divisible group. ::: :::{.theorem title="?"} For $\HH^3$, \[ P(\bd\HH^3) \iso P(\HH^3) \iso P(\bar \HH^3) ,\] so the Bloch-Wigner sequence deeply relates to all of these groups. ::: :::{.remark} The Bloch-Wigner dilogarithm can be realized as a map out of the Bloch group $P_\CC$. ::: > Links to hyperbolic 3-manifolds: ask Rodenko!