# Papers on discrete subgroups of Lie groups, deformation theory
and moduli spaces

## TWO FROM THE PAST

On the first Betti number of a hyperbolic manifold, Annals of Mathematics,
104 (1976), pp.235-24 7
A pdf version of this article is available

here.
Closed geodesics and the eta invariant, Annals of Mathematics, 108 (1978),pp.1-39.
A pdf version of this article is available

here.
## CONTROLLING DIFFERENTIAL GRADED LIE ALGEBRAS AND DEFORMATION THEORY

### With Dennis Johnson

Deformation Spaces Associated to Compact Hyperbolic Manifolds,
Papers in Honor of G.D. Mostow on His Sixtieth Birthday, edited by Roger Howe, Birkhauser, Progress in Mathematics; 67 ( 1986)
To obtain a copy of this paper in pdf format click

here
### P. Deligne's 1986 Letter on Controlling Differential Graded Lie Algebras

To obtain a copy of this letter in pdf format click
here
.
### With Bill Goldman

The Deformation Theory of Representations of Fundamental Groups of Compact
Kahler
Manifolds , Publ. Math. I.H.E.S., 67(1988),pp. 49-96.
To obtain a copy of this paper in pdf format click

here
.

### With Misha Kapovich

On Representation Varieties of 3-Manifold Groups
, Geometry and Topology, 21 (2017),pp. 1931-1968.
To obtain a copy of this paper in pdf format click

here
.

** **

**THE THETA CORRESPONDENCE AND SPECIAL CYCLES **

**
**
### With Steve Kudla

Intersection Theory of Cycles on Locally Symmetric Spaces and Fourier
Coefficients
of Holomorphic Modular Forms in Several Complex Variables,pp. 121-172.
Publ. Math. I.H.E.S., 71(1990)
To obtain a copy of this paper in pdf format click here
.
### With Jens Funke

Cycles with Local Coefficients for Orthogonal Groups and Vector-Valued
Siegel Modular Forms, American J. of Math., 128(2006), pp.899-948.
To obtain a copy of this paper in pdf format click

here
### With Jens Funke

Restrictions of Cohomological Theta Series for Orthogonal Groups
to the Borel-Serre Boundary, J. Inst. Math. Jussieu 12(2013), pp. 571-634.
To obtain a copy of this paper in pdf format click

here
### With Jens Funke

Spectacle Cycles with Coefficients and Modular Forms of Half Integral Weight, Arithmetic Geometry and
Automorphic forms, Volume in honor of the 60th birthday of Stephen S. Kudla, Advanced Lectures in Mathematics series.
International Press and the Higher Education Press of China, pp.91-154.
To obtain a copy of this paper in pdf format click
here
### With Jens Funke

The Geometric Theta Correspondence for Hilbert Modular Surfaces, Duke Math. J. 163 (2014), pp. 65-116.
To obtain a copy of this paper in pdf format click
here
### With Nicolas Bergeron and Colette Moeglin

Hodge Type Theorems for Arithmetic Manifolds, International Mathematical Research Notices
Associated to Orthogonal Groups,
submitted.
To obtain a copy of this paper in pdf format click
here
### With Nicolas Bergeron and Colette Moeglin

The Hodge Conjecture and Arithmetic Quotients
of Complex Balls, Acta Mathematica 216 (2016),pp.1 -125.
To obtain a copy of this paper in pdf format click
here
### With Nicolas Bergeron, Zhiyuan Li and Colette Moeglin

The Noether-Lefschetz Conjecture and Generalizations,
Inventiones Mathematicae 208 (2017), pp. 501-552.
To obtain a copy of this paper in pdf format click
here
## THE TOPOLOGY OF MODULI SPACES OF LINKAGES

### With Misha Kapovich

On the moduli space of polygons in the Euclidean plane
JDG 42 (1995),pp.430-464. A postscript
version of this article is available

here.
### With Misha Kapovich

Universality theorems for linkages and arrangements
Topology 35 (1996). A pdf version of this article is available

here.
## THE SYMPLECTIC AND TORIC GEOMETRY OF MODULI SPACES OF N-GON LINKAGES

( In the three-dimensional model spaces of constant curvature)

### With Misha Kapovich

The symplectic geometry of polygons in Euclidean space
JDG. 44 (1996). A pdf version of this article
is available

here.
### With Ben Howard and Chris Manon

The toric geometry of triangulated polygons in Euclidean space
Canadian J. of Math. (2011)). A pdf version of this article
is available

here
### With Misha Kapovich and Tom Treloar

The symplectic geometry of polygons in hyperbolic 3-space
.
Asian J. of Math. 4 (2000), (papers dedicated to the memory of
K. Kodaira). A pdf version of this article is available

here.
### By Tom Treloar

The symplectic geometry of polygons in the 3-sphere (by T.Treloar)
Canadian J. of Math. 54 (2002).
A pdf version of this article is available

here.
(In the space of positive definite Hermitian matrices)

### With Hermann Flaschka

The moduli space of weighted configurations on projective space (with H.
Flaschka). Canadian J. of Math. 57(2005). A pdf version of this article is available

here.
## THE GENERALIZED TRIANGLE INEQUALITIES IN SYMMETRIC SPACES AND
EUCLIDEAN BUILDINGS
AND THE CONNECTION WITH THE STRUCTURE CONSTANTS OF REPRESENTATION RINGS
AND HECKE ALGEBRAS

### With Misha Kapovich and Bernhard Leeb

Convex functions on symmetric spaces and geometric invariant theory for
spaces of weighted configurations
on flag manifolds, Journal of Diff, Geom. 81(2009)
A pdf version of this
article is available here.
### With Misha Kapovich and Bernhard Leeb

Polygons in symmetric spaces and Euclidean buildings, Geometry and Functional Analysis 19(2009).
A pdf version of this
article is available here.
### With Misha Kapovich and Bernhard Leeb

The generalized triangle inequalities in symmetric spaces and buildings
with applications to algebra ,
Memoirs of the AMS A pdf version of this
article is available
here.
### With Misha Kapovich and Bernhard Leeb

A path model for geodesics in Euclidean buildings and its applications
to representation theory,
A pdf version of this
article is available
here.
### With Misha Kapovich and Shrawan Kumar

Saturation and irredundancy for Spin(8)
. A pdf version of this article is available
here.
### Misha Kapovich's Madrid ICM talk (2006)

A pdf version of this
article is available
here.
## DEFORMATIONS
OF REPRESENTATIONS OF FINITELY GENERATED GROUPS

### With Misha Kapovich

On the deformation theory of representations of fundamental groups of compact
hyperbolic 3-manifolds,
Topology 35 (1996). A pdf version
of this articles is available here. .
### With Misha Kapovich

On representation varieties of Artin groups,projective arrangements and
the fundamental groups of smooth algebraic varieties, (with M. Kapovich),
Publ. Math. IHES, 88 (1998).
A pdf version of this article
is available here.
## FLAT
CONNECTIONS AND REPRESENTATIONS OF GENERALIZED ARTIN GROUPS OF LIE TYPE

### With Misha Kapovich

Quantization of bending deformations of polygons in Euclidean 3-space,
hypergeometric integrals and the Gassner representation (with M.Kapovich),
Canad. Math. Bull. 44 (2001). A pdf version of this article
is available here.
### With Valerio Toledano

Casimir operators and monodromy representations of generalized braid groups,
Transformation Groups, 10, 2005.
A pdf version of this article is available
here .
## THE
PROJECTIVE INVARIANTS OF ORDERED POINTS ON THE LINE

Over the last four years I have been engaged in a collaboration with
Ben Howard, Andrew Snowden and Ravi Vakil. This HMSV collaboration
has resulted in six papers culminating with the paper below
"

*The ideal of relations for the moduli space of n points on the line*".
Our first paper computed

*the equations of the moduli space*. It
was published in Duke Math J., Vol. 46, 2009, 175-226.

### Ben Howard, Andrew Snowden and Ravi Vakil

The equations for the moduli space of n points on the line,
A pdf version of this article is available
here.
We have just solved the much harder problem of finding the

* relations between the generating invariants* found by Kempe
in 1894. It will appear in the Journal of the European Mathematical Society.

### With Ben Howard, Andrew Snowden and Ravi Vakil

The ideal of relations for the moduli space of n points on the line
(with ).
A pdf version of this article is available
here.
The Comptes Rendus announcement of the previous paper is
here.
We refined the above to show that the above presentation of the ring of projective invariants (proved above over the rational numbers) hold
over the ring Z[1/12]. It will appear in Communications in Algebra.

### With Ben Howard, Andrew Snowden and Ravi Vakil

The ideal of relations for the moduli space of n points on the line: integrality results
A pdf version of this article is available here
The hard part of the proof is to reduce to the eight-point case which we solved in
the paper
### With Ben Howard, Andrew Snowden and Ravi Vakil

The ring of projective invariants of eight points on the line via representation theory, to
appear in the Proceedings of the London Math. Soc.
A pdf version of this article is available here