Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups - MathOverflow

Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups

2012-07-21T00:15:04Z

Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)$. If one considers the group scheme $\mathfrak G$ associated to $G$, there is a nice argument that shows that infinitesimal deformations of a homomorphism $\phi\in\mathfrak R(\mathbb R)$ are the (Zariski) tangent space to $\mathfrak R$ at $\phi$ given by the kernel of the map $$ \mathfrak R(\eta)\colon\mathfrak R(\mathbb R[\epsilon])\to \mathfrak R(\mathbb R) $$ where $\mathbb R[\epsilon]$ is the algebra of dual numbers and $\eta$ is the augmentation $\eta(\epsilon)=0$.

One checks that $$\mathfrak R(\mathbb R[\epsilon])=\mathrm{Hom}(\Gamma,\mathfrak G(\mathbb R[\epsilon]))=\mathrm{Hom}(\Gamma,\mathfrak g\rtimes G) $$ and $T_\phi(\mathfrak R)$ is the fibre of $\mathfrak R(\eta)$ over $\phi$, i.e. homomorphisms into $\mathfrak g\rtimes G$ that project onto $\phi$, which must be of the form $\gamma\mapsto(\tau(\gamma),\phi(\gamma))$ for some function $\tau\colon\Gamma\to\mathfrak g$. Finally, the requirement that this homomorphism be indeed a homomorphism implies that $\tau$ must be a $1$-cocycle, i.e. $\tau\in Z^1(\Gamma,\mathfrak g)$, where $\Gamma$ acts on $\mathfrak g$ via $\phi\colon\Gamma\to G$.

My question is, whether the above is also true for Lie groups, which in general may not be algebraic groups.

Answer by paul garrett for Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups

2012-07-21T00:41:44Z

A quick small answer: for reductive real Lie groups, there are the "rigidity theorems" of Mostow, Margulis, and others. Basically, the point is that for real rank above 1, "lattice subgroups" admit no deformations whatsoever (and are essentially "arithmetic"). These groups are typically either algebraic or isogenous to an algebraic group (the latter illustrated significantly by "metaplectic" two-fold covers of symplectic groups Sp(n,R).

I do not know about nilpotent and solvable Lie groups, which I do suspect are less algebraic (already illustrated by various incarnations of three-dimensional Heisenberg groups, within my ken).

Answer by Misha for Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups

2012-08-18T05:18:07Z

This fact holds for general Lie groups $G$ (which I will equip with a real-analytic structure) and finitely-generated groups $\Gamma$. It is explained in detail in Raghunathan's book "Discrete subgroups of Lie groups", sections 6.1-6.9), but goes back to Andre Weil's paper "Remarks on cohomology of groups", Annals of Math, 1964. Here is an outline:

Let $x_1,...,x_n$ be generators of $\Gamma$ and $R_i, i\in I$, be the relators (this set need not be finite). Each relator $R_i$ (regarded as a word in the letters $x_k^{\pm 1}$) defines an analytic map $f_i: G^n\to G$. Thus, $Hom(\Gamma,G)$ is identified naturally with the real-analytic variety $$ {(g_1,...,g_n)\in G^n: f_i(g_1,...,g_n)=1, i\in I}, $$ $$ \rho\mapsto (\rho(x_1),...,\rho(x_n)). $$ Now, you say that for $\rho\in Hom(\Gamma,G)$, the tangent space $T_{\rho}(\Gamma,G)$ is the intersection of kernels of the derivatives $df_i$ at the point $\rho$. In the case when $G$ is algebraic, this is the Zariski tangent space, otherwise, this is the tangent space $m_\rho/m^2_\rho$ of $Hom(\Gamma,G)$ regarded as an analytic variety (that need not be reduced). Weil and Raghunathan also explain why this is natural in the context of group homomorphisms. Then, they prove that $T_{\rho}(\Gamma,G)\cong Z^1(\Gamma, Ad\circ \rho)$, see 6.9 in Raghunathan's book (he is only interested in local rigidity, but his proof is general). In particular, if $H^1(\Gamma, Ad\circ \rho)=0$, then $\rho$ is locally rigid.

The advantage in the algebraic groups case is that you do not need to go through group presentations to see all this, maybe there is a similarly clean proof in the real analytic case.