Articles

# What is a differential form?

There are all sorts of answers to that question in the title, such as

• it’s an alternating tensor
• it’s a section of an exterior power of the cotangent bundle
(in particular, differential 1-forms are the dual objects to vector fields)
• it’s something that can be integrated

but none of these seem very satisfying to me in and of themselves—they give you the technical specifications to formally define things, but leave the motivation for defining them somewhat of a mystery (although the last answer kind of hints at some of this motivation.) Imagine you were trying to figure out what a vacuum cleaner was, and someone told you “they’re air pumps which are set up to create partial vacuums; you can roll them along your carpet.”

With that in mind, let me supplement the above answers with an additional, rather vague but hopefully slightly more motivational one: in the words of an (unidentified) analysis postdoc at Chicago via Alan Chang, differential (2-)forms are “pretentious parallelograms” (and I suppose more general k-forms would be pretentious parallelopipeds.)

In other words, they are compact book-keeping devices which help keep track of the data needed to make sense of things like integration, or solutions to differential equations, across coordinate charts, in a coordinate-free way—meaning, for the most part, that the coordinates are there waiting to be unrolled if you need them, but stay hidden, out the way and unobtrusive, as long as you don’t.

### For example

Consider integrating a 3-form $\omega$ over a 3-manifold. In order for such an integration to be globally well-defined, we would want overlap transformations between coordinate charts U, V to satisfy $\omega_V = (\varphi_{UV})_*(\omega_U)$, where $(\varphi_{UV})_*$ denotes the pullback operator.

We may verify (i.e. it is a computation to verify) that choosing $\omega$ to be a (smooth) section of the exterior product vector bundle $\bigwedge^3 T^*M$ satisfies this.

A longer discussion of why exterior products might come into play may be found in Section 5.1 of Simon Donaldson’s excellent Riemann Surfaces.

To go back to and slightly overstretch the analogy with the vacuum cleaner—now that we’ve been told that the vacuum cleaner is meant to help clean your carpet, this computation would be analogous to figuring out why using an air pump to create a partial vacuum and then rolling that vacuum around your carpet might achieve that objective.

### Okay, what about a (-1,1)-form?

Looking at differential equations of various shapes, and/or complex structures and various regularity conditions gives rise to varying transformation rules, which leads to the minting of such related exoticisms as (-1,1)-forms—also known as (after imposing a few more technical conditions) Beltrami differentials—and holomorphic quadratic differentials, both of which play an important role in Teichmüller theory.

These varying exoticisms may also be viewed as sections of suitable vector bundles: e.g. a (-1,1)-form is a section of the tensor product $T\Sigma \otimes \overline{T^*\Sigma}$, where $T\Sigma$ denotes the (holomorphic) tangent bundle (canonically identified with its double dual) and $T^*\Sigma$ its dual.

### Why would I ever need a (-1,1)-form?

Beltrami differentials are naturally obtained as (essentially bounded) solutions $\mu$ to the differential equation $\dfrac{\partial w}{\partial \overline{z}} = \mu \dfrac{\partial w}{\partial z}$ where w is a given complex distribution (i.e. generalized function) of z with locally $L^2$ derivative.

Such an equation (known as the Beltrami equation) comes up e.g. when one seeks to find isothermal coordinates on a Riemannian surface, i.e. coordinates in which the given metric is conformal to a Euclidean one: if the given metric is given by $ds^{2}=E\,dx^{2}+2F\,dxdy+G\,dy^{2}$, then we may rewrite it, using the complex coordinate $z = x + iy$, as $ds^2 = \lambda|dz + \mu d\bar{z}|^2$, for suitable $\lambda$ and $\mu$, which turn out to be smooth and satisfy $\lambda > 0$ and $|\mu| < 1$ (explicit formulae may be found here.)

In isothermal coordinates (u, v) we would have $ds^{2}= \rho (du^{2}+dv^{2})$ with $\rho > 0$ smooth; note the complex coordinate $w = u + iv$ satisfies $|dw|^2 |w_z|^2|\,dz + \frac{w_{\bar{z}}}{w_z} \,d\bar{z}|^2$.

Since $ds^2 = \rho |dw|^2 = |dw|^2 |w_{z}|^2 |\,dz + \frac{w_{\bar{z}}}{w_z} \,d{\bar{z}}|^2 = \lambda|dz + \mu d\bar{z}|^2$, we will be done if $\rho|w_z|^2 = \lambda$ and $\mu = \dfrac{w_{\bar{z}}}{w_z}$. The former is no constraint, since we are free to choose $\latex rho$ to be any positive smooth function; hence we are left with the latter, which is exactly the Beltrami equation with $\mu$ given and where we wish to solve for w.

In other words, there is a bijective correspondence between conformal structures on a Riemann surface and Beltrami differentials.

Beltrami differentials are also closely related to quasiconformal homeomorphisms via the measurable Riemann mapping theorem, and thus appear in Teichmüller theory, as e.g. tangent vectors to Teichmüller space.

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# Ergodicity of the Geodesic Flow

### The geodesic flow

Given any Riemannian manifold M, we may define a geodesic flow $\varphi_t$ on the unit tangent bundle $T^1M$ which sends a point (x, v) to the point $(\varphi_t x, \varphi_t^* v)$, where

• $\varphi_t x$ is the point distance from x along the geodesic ray emanating from x in the direction of v, and
• $\varphi_t^* v$ is the parallel transport of v along the same ray

(it’s a mouthful, isn’t it? It’s really simpler than all those words make it seem.) Note, at each point, we remember not just where we are—the point $x \in M$—, but also where we’re going—the direction vector $v \in T_x M$; if we were to forget this second piece of information, we would become a little unmoored: here we are … where should we go next?

### Ergodicity

When M is a closed (i.e. compact, no boundary) hyperbolic surface, or more generally closed with strictly negative curvature, this geodesic flow is ergodic, i.e. any subset of $\Sigma$ or M invariant under the flow has either zero measure, or full measure. Here the measure on our Riemannian manifold is the pushforward of the Lebesgue measure through the coordinate charts.

Since linear combinations of step functions are dense in the space of bounded measurable functions, we may equivalently define ergodicity as: any measurable function invariant under the flow is a.e. constant.

(Side note: with more assumptions on the curvature we may relax the compactness assumption to a finite volume assumption)

### The Hopf argument (for closed hyperbolic manifolds)

This is essentially due to the exponential divergence of geodesics in negative curvature , and the splitting of the tangent spaces $T_ v T^1M = E^s_v \oplus E^0_v \oplus E^u_v$ into stable, tangent (flowline), and unstable distributions; these give rise to three maximally transverse foliations, the stable foliation $W^s$, the unstable foliation $W^u$, and the foliation by flowlines $W^0$.

The flow is exponentially contracting in the forward time direction on the leaves of the stable foliation $W^s$, and on which the flow is exponentially contracting in the reverse time direction the leaves of the unstable foliation $W^u$. In other words, the flow is Anosov.

We may describe these foliations explicitly in the case of constant negative curvature—if we take $\gamma$ to be the geodesic tangent to $v \in T^1M$,

• $W^s$(v) is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v) \in M$ tangent to the forward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$. “forward” here being taken with reference to how v is pointing along $\gamma$;
• $W^u(v)$ is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v)$ tangent to the backward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$;
• $W^0(v) = \gamma$.

#### Step 1

Suppose f is a $\phi$-invariant function; by replacing f with min(f, C) if needed, WMA f is bounded. Since continuous functions are dense in the set of measurable functions on M, we may approximate f in $L^1$ by bounded continuous functions $h_\epsilon$.

By the Birkhoff ergodic theorem, forward time averages [w.r.t. $\phi$] exist for $h_\epsilon$.

By an argument involving the $\phi$-invariance of f and the triangle inequality, f is well-approximated (in $L^1$) by the forward time averages of $h_\epsilon$.

#### Step 2

The forward time averages of $h_\epsilon$ are constant a.e., since by invariance these averages are already constant a.e. on (each of) the leaves of $W^0$, and they are also constant a.e. on (each of the) unstable and stable leaves, by uniform continuity of $h_\epsilon$.

#### Step 3

To conclude that time averages, and hence our original arbitrary integrable function, are constant a.e. on M, we (would like to!) use Fubini’s theorem: locally near each $(x_0,v_0) \in T^1M$, the set of (x, v) along each of the foliation directions at which the time averages are equal to those at $(x_0,v_0)$ has full measure, by the previous Step.

By Fubini’s theorem applied to the three foliation directions, we (would) conclude that the set of nearby (x, v) at which the time averages are equal to those at $(x_0,v_0)$ has full measure. Hence the time averages are locally constant, and since $T^1M$ is connected we are done.

### But! (Also more generally, for K < 0)

The problem is that while our stable and unstable leaves are differentiable, the foliations need not be—i.e. the leaves may not vary smoothly in their parameter space.

To justify the use of a Fubini-type argument one instead shows that that these foliations are absolutely continuous.

The proof then immediately generalizes to all compact manifolds with (not necessarily constant) negative sectional curvature. For more general negatively-curved nanifolds, the stable and unstable foliations $W^s$ and $W^u$ may still be described in terms of unit normal bundles over horospheres, where horospheres are now described, more generally, as level sets of Busemann functions.

The proof of absolute continuity of the foliations proceeds as follows

1. Showing that the stable and unstable distributions $E^s$ and $E^u$ (also the “central un/stable” or “weak un/stable” distributions, i.e. $E^{s0} := E^s \oplus E^0$ and  $E^{u0} := E^u \oplus E^0$) (of any $C^2$ Anosov flow) are Hölder continuous—i.e. given $x, y \in M$, the Hausdorff distance in $TTM$ between the stable subspace $E^s(x)$ and the stable subspace $E^s(y)$ is $\leq A \cdot d(x,y)^\alpha$.
Roughly speaking, this is true because any complementary subspace to $E^s$ will become exponentially close to $E^s$ under the repeated action of the geodesic flow, by the same mechanism that makes power iteration tick; and the distance function on M is Lipschitz. Analyzing the situation more carefully, and applying a bunch of simplifying tricks such as the adjusted metric described in Brin’s section 4.3, yields the desired Hölder continuity.
2. Using this, together with the description of horospheres as limits of sequences of spheres with radii increasing to $+\infty$, to establish that between any pair of transversals for the un/stable foliation, we have a homeomorphism which is $C^1$ with bounded Jacobians, and hence absolutely continuous.
Very slightly less vaguely, Hölder continuity of $E^{u0}$, together with the power iteration argument as above, implies tangents to transversals to the stable foliation $W^s$ become exponentially close; given regularity of the Riemannian metric, this implies the Jacobians of the iterated geodesic flow on these transversals become exponentially close. By a chain rule argument and another application of the power iteration argument, this implies that the Jacobians of the map between transversals are bounded.
This condition on the foliations is known as transversal absolute continuity, and implies, by a general measure theoretic argument (see section 3 of Brin’s article), absolute continuity of the foliations.
3. Note that this last step, at least as presented in Brin, appears to require the use of pinched negative curvature.

### References

Eberhard Hopf, “Ergodic theory and the geodesic flow on surfaces of constant negative curvature.” Bull. Amer. Math. Soc. 77 (1971), no. 6, 863–877.

Yves Coudene, “The Hopf argument.

Misha Brin, “Ergodicity of the Geodesic Flow.” Appendix to Werner Ballman’s Lectures on Spaces of Nonpositive Curvature.

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Overview / Outlines

# (Some) Lie theory for geometric topology / dynamics

(following an outline by Wouter van Limbeek; filling this outline is a work-in-progress. Sections 6 and 7 are particularly incomplete; sections 11 and 12—or 8 onwards—could [should?] be split to form its own post.)

Here we are working with connected Lie groups.

1) Classes of Lie groups: abelian, nilpotent, solvable, semisimple

Abelian Lie groups are completely classified: these all have the form $G \cong \mathbb{R}^k \times \mathbb{T}^e$ (where $\mathbb{T}^e$ denotes the e-dimensional torus.)

Nilpotent groups are those with a terminating lower central series $G_{(k+1)} = [G, G_{(k)}]$. Examples of nilpotent Lie groups: Heisenberg groups, unitriangular groups, the (3-by-3) Heisenberg group mod its center.

Solvable groups are those with a terminating upper central series $G^{(k+1)} = [G^{(k)}, G^{(k)}]$. There is an equivalent formulation in terms of composition series—in this sense solvable groups are built up from (simple) abelian groups. Examples of solvable Lie groups:

• $\mathrm{Sol} = \mathbb{R}^2 \rtimes \mathbb{R}$, where $t \in \mathbb{R}$ acts on $\mathbb{R}^2$ as the matrix $\left( \begin{array}{cc} e^t \\ & e^{-t} \end{array} \right)$.
• $\mathrm{Aff}(\mathbb{R})^2 \cong \mathbb{R} \rtimes \mathbb{R}_{> 0}$, where the action is by scalar multiplication.
• The group of all upper-triangular matrices.
• $\mathbb{R}^2 \rtimes \mathbb{R}$, lifted from $\mathbb{R}^2 \rtimes \mathrm{SO}(2)$, where the action is by rotation.

By definition, we have abelian $\subset$ nilpotent $\subset$ solvable

Simplicity and semisimplicity

A Lie algebra $\mathfrak{g}$ is simple if it has no proper ideals (i.e. proper subspaces closed under the Lie bracket.) $\mathfrak{g}$ is semisimple if it is a direct sum of simple Lie algebras.

A Lie group G is (semi)simple if its Lie algebra $\mathfrak{g}$ is (semi)simple.

Equivalently, G is simple if it has no nontrivial connected normal subgroups (what can go wrong if we don’t have that additional adjective?)

G is semisimple if its universal cover $\tilde{G}$ is a direct product of simple Lie groups. (Note that things can go wrong if we do not take the universal cover, e.g. the quotient of $\mathbb{SL}_2(\mathbb{R})$ by $\mathbb{Z}/2\mathbb{Z}$, where the action is the diagonalisation of $\pm 1 \curvearrowright \mathbb{SL}_2(\mathbb{R})$, is semisimple [in the sense that its Lie algebra is semisimple], but not a product.)

Relation between these types and the adjoint representation.

Recall the adjoint representation $\mathrm{Ad}: G \to \mathrm{GL}(\mathfrak{g})$ given by sending a group element g to the matrix representing $D_e c_g$, where $c_g$ denotes conjugation by g. Now

• G is abelian iff Ad(G) is the trivial representation
• G is nilpotent iff Ad(G) is unipotent
• G is solvable iff Ad(G) is (simultaneously) triangularizable (over an algebraically-closed field)
• G is semisimple iff Ad(G) is a semisimple representation (i.e. may be written as a direct sum of irreducible Ad-invariant representations.)
• G contains a lattice (see below) iff Ad(G) $\subset \mathrm{SL}(\mathfrak{g})$.

2) Levi decomposition of Lie groups

Note that subgroups of solvable groups are solvable, and extensions of solvable groups by solvable groups are solvable. Putting all of these together, we obtain that any Lie group G has a unique maximal closed connected normal solvable subgroup R (called the solvable radical of G.)

The Levi decomposition (also called, in some contexts, the Levi-Malcev decomposition) of G is G = RS, where R is the solvable radical, $S \subset G$ is semisimple, and $R \cap S$ is discrete.

If G is simply-connected, then $G = R \rtimes S$.

3) Classification of compact Lie groups

Any compact connected solvable Lie group G is a torus (Proof sketch: Induct on length of upper central series. For n=1, G is abelian and we are done. Otherwise, we have the short exact sequence $1 \to [G,G] \to G \to \frac{G}{[G,G]} \to 1$; the outer terms are tori $T_1$ and $T_2$, and we have a map $T_2 \to \mathrm{Out}(T_1) = \mathrm{GL}(\dim T_1, \mathbb{Z})$. Since this is a group homomorphism from a connected group to a discrete group, it musth ave trivial image; hence $T_1 \subset G$ is central, and we can induce on $\dim T_2$ to produce a section $\frac{G}{[G,G]} \to G$.

Compact connected semisimple Lie groups can also be classified, but this story involves much more algebra (representation theory, highest weights, etc.

Using the Levi decomposition, we have the following

Theorem. Any compact connected Lie group G is isomorphic to $(\mathbb{T}^k \times G_1 \times \dots \times G_n) / F$, where F is a finite group and $G_1, \dots, G)n$ are simple.

4) When is a Lie group an algebraic group? When is a Lie group linear?

An algebraic group is a group with a compatible algebraic variety structure (i.e. multiplication and inversion are regular maps.) A linear group is (isomorphic to) a subgroup of GL(n,k) for some field k. Note GL(n,k) is algebraic, and thus linear groups defined by polynomial equations (but not all linear groups!) are algebraic.

A Lie group is algebraic if it is isomorphic to a linear algebraic group (but this is not an iff [?])

Many of the classical Lie groups are (linear) algebraic groups, but note that not all Lie groups are algebraic: e.g. $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ is not algebraic because its center is not algebraic (“is too large to be algebraic.”)

5) When is the exponential map a diffeomorphism?

We have an exponential map $\exp: \mathfrak{g} \to G$. Its derivative at the identity, $\exp_*: \mathfrak{g} \to \mathfrak{g}$, is the identity map; exp therefore restricts to a diffeomorphism from some neighborhood of 0 in $\mathfrak{g}$ to a neighborhood of 1 in G.

If G is connected, simply-connected, and nilpotent, the exponential map exp is a (global) diffeomorphism (in fact, an analytic isomorphism of analytic manifolds, if G is linear algebraic.)

6) Topology of Lie groups: fundamental group, homotopy type, cohomology …

Theorem. Any connected Lie group has abelian fundamental group.

Proof sketch: In fact this is true for any connected topological group, because the group structure forces things to be nice that way. To make this precise can be mildly annoying though.

Theorem (Weyl). The fundamental group of a compact semisimple Lie group is finite

Theorem. Any connected Lie group has trivial second homotopy and torsionfree third homotopy.

Proof sketch: Since any Lie group retracts onto its maximal compact subgroup, WLOG we work with only connected groups.

From the long exact sequence of homotopy groups obtained from the path-loop fibration, $\pi_k(G) \cong \pi_{k-1}(\Omega G)$. By Morse theory, $\pi_1(\Omega G) = 0$ and $\pi_2(\Omega_G) = \mathbb{Z}^t$.

Many more things can be said: see e.g. this survey.

Relation with the Lie algebra

7) Geometry of Lie groups: relation between geometry of an invariant metric and algebraic structure.

Note that a Lie group structure yields a natural left-invariant (or right-invariant) metric, given by propagating the inner product at the identity by group multiplication. With additional hypotheses, we can make this bi-invariant:

Theorem. Every compact Lie group G admits a bi-invariant metric.

Idea of proof Use the natural Haar measure on G to average the left-invariant metric.

We can play this invariant metric and the group structure off against each other: e.g. the geodesics of G coincide with the left-translates of 1-parameter subgroups of (at the identity); as a corollary of this, we obtain that any Lie group G is (geodesically) complete.

8) Selberg’s lemma

Lemma (Selberg 1960). A finitely generated linear group over a field of zero characteristic is virtually torsion-free.
(cf. Theorem (Malcev 1940). A finitely generated linear group is residually finite.)

Proof: Using number fields (local fields, see Cassels or Ratcliffe), or Platonov’s theorem, which seems to be a bunch of commutative algebra (see Bogdan Nica’s paper.)

9) Finite generation/presentation of lattices: Milnor-Svarc and Borel-Serre.

A lattice is a discrete group $\Lambda \leq G$ with $G / \Lambda$ of finite volume (as measured by the natural Haar measure on G.) The prototypical example is $\mathrm{SL}(2,\mathbb{Z}) \subset \mathrm{SL}(2, \mathbb{R})$, which of course is the isometry group of the torus.

Facts

1. Any lattice in a solvable group is co-compact (also called uniform.)
2. (Borel, Harish-Chandra) Any noncompact semisimple group contains a lattice that is not co-compact (also called non-uniform.)

Theorem (Milnor-Svarc). If $\Gamma$ acts on a proper geodesic metric space X properly discontinuously and cocompactly, then $\Gamma$ is finitely-generated.

Proof: Since $\Gamma \curvearrowright X$ cocompactly, the action has a compact fundamental domain $Z = X / \Gamma$. By the proper discontinuity of the action, the vertices of Z cannot accumulate. But compactness then implies Z has finitely many vertices, and hence finitely many sides. Since each algebraically independent generator would add a side to Z, this implies that $\Gamma$ is finitely generated.

In fact, many non-uniform lattices are also finitely-generated.

The $\mathbb{R}$-rank of a Lie group G is the dimension of the largest abelian subgroup of Z(a) simultaneously diagonalizable over $\mathbb{R}$, as a varies over all semisimple elements of G ($a \in G$ is semisimple if Ad(a) is diagonalizable over $\mathbb{R}$. Geometrically, we may interpret the $\mathbb{R}$-rank as the dimension of the largest flat in G.

A locally-compact group G is said to have property (T) if for every continuous unitary representation $\rho: G \to \mathcal{U}(\mathcal{H})$ into some Hilbert space there exists $\epsilon > 0$ and compact $L \subset G$ s.t. if $\exists v \in \mathcal{H}$ with $\|v\| = 1$ s.t. $\|\rho(l) v - v\| < \epsilon$ for all $l \in L$, then $\exists v' \in \mathcal{H}$ with $\|v'\| = 1$ s.t. $\rho(G)$ fixes v‘. (i.e., the existence of an almost-invariant vector implies the existence of a fixed point.)

Theorem (Kazhdan, 1968): If G is a simple Lie group of $\mathbb{R}$-rank $\geq 2$, and $\Gamma \subset G$ is a lattice, then $\Gamma$ has property (T), and hence is finitely-generated.

Alternative proof: every such lattice is arithmetic, then use a fundamental domain and argue [as before]

10) Levi-Malcev decomposition of lattices

The Levi-Malcev decomposition as applied to a lattice $\Lambda \subset G = RS$ tells us that $\Lambda = \Psi \Sigma$ where $\Psi = \Lambda \cap R$ is a lattice in a solvable group and $\Sigma = \Lambda \cap S$ is a lattice in a semisimple group: note that each of these intersections remains discrete, and has finite co-volume for if not $\Lambda$ would not have finite co-volume.

Hence the general scheme for understanding lattices in Lie groups: first understand lattices in solvable groups and in semisimple groups, then piece them together using the Levi-Malcev decomposition …

11) (Non-)arithmetic lattices. When is a lattice arithmetic?

Most generally, an arithmetic subgroup of a linear algebraic group G defined over a number field K is a subgroup Γ of G(K) that is commensurable with $G(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers of K.

Hence we may define more abstractly a lattice $\Gamma$ in a connected (solvable) Lie group G is said to be arithmetic if there exists a cocompact faithful representation i of G into $G^*_{\mathbb{R}}$ (where $G^* \subset \mathrm{GL}(n, \mathbb{C})$ is an algebraic subgroup) with closed image and s.t. $i(\Gamma) \cap G^*_{\mathbb{Z}}$ has finite index in both $\Gamma$ and $G^*_{\mathbb{Z}}$.

We have that

Theorem (Borel, Harish-Chandra). Arithmetic subgroups are lattices.

Conversely,

Theorem (Mostow). (4.34 in Raghunathan.) Let G be a simply-connected solvable Lie group and $\Gamma \subset G$ a lattice. Then G admits a faithful representation into $\mathrm{GL}(n, \mathbb{R})$ which sends $\Gamma$ into $\mathrm{GL}(n, \mathbb{Z})$.

Note, however, the counterexample on pp. 76-77 of Raghunathan.

Much stronger results are true in semisimple Lie groups:

Theorem (Margulis arithmeticity). Any irreducible lattice in a semisimple Lie group with no rank one factors is arithmetic.

(More precisely, see statement of Selberg’s conjectures in Section 7 of this article.)

Theorem (Margulis’ commensurator criterion.) Let $\Gamma < G$ be an irreducible lattice (where G may have rank 1). Then $\Gamma$ is arithmetic iff the commensurator of $\Gamma$ is dense in G.

Recall the commensurator of $\Gamma < G$ is the subset of G consisting of elements g s.t. $\Gamma$ and $g \Gamma g^{-1}$ are commensurable

12) Rigidity of lattices

(i.e. when can you deform a lattice in the ambient Lie group?)

Theorem (Weil local rigidity). Let $\Gamma$ be a finitely generated group, G a Lie group and $\pi: \Gamma \to G$ be a homomorphism. Then $\Gamma$ is locally rigid if $H^1(\Gamma, \mathfrak{g}) = 0$. Here $\mathfrak{g}$ is the Lie algebra of G and $\Gamma$ acts on $\mathfrak{g}$ by $\mathrm{Ad}_G \circ \pi$.

Theorem (Mostow rigidity). Let Γ and Δ be discrete subgroups of the isometry group of $\mathbb{H}^n$ (with n > 2) whose quotients $\mathbb{H}/\Gamma$ and $\mathbb{H}/\Delta$ have finite volume. If Γ and Δ are isomorphic as discrete groups, then they are conjugate.

i.e. lattices in hyperbolic space are pretty darn rigid. But then there’s even more. A lattice is caled irreducible if no finite index subgroup is a product;

Theorem (Margulis superrigidity). Let Γ be an irreducible lattice in a connected semisimple Lie group G of rank at least 2, with trivial center, and without compact factors. Suppose k is a local field.

Then any homomorphism π of Γ into a noncompact k-simple group over k with Zariski dense image either has precompact image, or extends to a homomorphism of the ambient groups.

Remark. Margulis superrigidity implies Margulis arithmeticity (how?)

Margulis’ normal subgroup theorem. Let G be a connected semisimple Lie group of rank > 1 with finite center, and let $\Gamma < G$ be an irreducible lattice.

If $N \leq \Gamma$ is a normal subgroup of $\Gamma$, then either N lies in the center of G (and hence $\Gamma$ is finite), or the quotient $\Gamma / N$ is finite.

Quasi-isometric rigidty of lattices

Zariski tangent space to representation variety and cohomology.

References

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Annotated Definitions

# The Cross-ratio

The cross-ratio of four points is $[z_1, z_2, z_3, z_4] := \frac{z_1 - z_3}{z_1 - z_4} \frac{z_2 - z_4}{z_2 - z_3}$.

By taking $z_1, z_2, z_3, z_4 \in \mathbb{R}$ we define the cross-ratio on (ordered) quadruples of points on the real line, and thereby, by interpreting the differences as signed distances, on ordered quadruples of collinear points in Euclidean space. We can extend this definition to the projective line or projective space by allowing $z_i = \infty$.

By taking $z_1, z_2, z_3, z_4 \in \mathbb{C} \cup \{\infty\}$, we define the cross-ratio on quadruples of points in the Riemann sphere $\hat{\mathbb{C}}$. (More generally, we can define the cross-ratio on quadruples of points in any field.)

The cross-ratio is interesting because it is the essential projective invariant (in 2 dimensions); indeed, this might be taken as the more “natural” or “universal” definition, or what Tim Gowers refers to when he says that [about tensor products in particular, but also mathematical objects in general] “exactly how they are defined is not important: what matters is the properties they have.”

More precisely, the cross-ratio is invariant under projective transformations, i.e. (in 2 dimensions) under $\mathrm{Aut}(\hat{\mathbb{C}}) \cong \mathrm{PSL}(2, \mathbb{C}) := \mathrm{SL}(2, \mathbb{C}) / \pm I$. It is essentially the only projective invariant of ordered quadruples of points, in the sense of a universal property (any projective invariant can be bijectively transformed into the cross-ratio), since $\mathrm{PSL}(2, \mathbb{C})$ acts simply transitively on ordered triples of points in $\hat{\mathbb{C}}$. [I feel like I don’t quite understand the details of the argument here yet.]

Furthermore, quadruples of points are the natural choice of objects to  determine (2-8dimensional) projective invariants on: the Euclidean distance between 2 points is invariant under translations or rotations and the ratio of the distances between 3 points is invariant under Euclidean similarities, but $\mathrm{PSL}(2, \mathbb{C})$ acts transitively on ordered triples of points in $\hat{\mathbb{C}}$.

This property of invariance of cross-ratio allows us to define hyperbolic distance in terms of the cross-ratio: more precisely, taking the Poincaré half-plane model $\mathbb{H}$ of the hyperbolic plane, we have that $d_{\mathbb{H}}(x,y) = \log [x', y, x, y']$ where $x', y'$ are the endpoints on $\partial\mathbb{H}$ of the geodesic between $x$ and $y$ (we may replace $\mathbb{H}$ with the Poincaré disc model $\mathbb{D}$ throughout.)

To prove this we conformally map our points to the imaginary axis and treat only this special case: note that $\mathrm{Aut} (\mathbb{H}) \subset \mathrm{Aut}(\widehat{\mathbb{C}})$ and also that $\mathrm{Aut} (\mathbb{H})$ preserves the cross-ratio. Now WLOG let $x, y$ lie on the imaginary axis with $x = i$ and $y = ai$ (where $a \in \mathbb{R}$.) Then $d_{\mathbb{H}}(x,y) = a$ and $[x',y,x,y'] = \frac{0-ai}{0-i} \frac{\infty - i}{\infty-ai} = a$, as desired.

Standard