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Daniel Ramras

Publications and source records attributed to Daniel Ramras.

7 recordsLinked to original sources
arXiv2019

Bad Representations and Homotopy of Character Varieties

Let G be a connected reductive complex affine algebraic group, and let X denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in X, extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of X, proving conjectures of Florentino-Lawton-Ramras. Lastly, using the earlier analysis of Borel-de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.

math.AG↗
arXiv2017

Wonderful Compactification of Character Varieties

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when G=PGL(n,C). Additionally, when F is a free group we identify the boundary divisors of Y in terms of previously studied moduli spaces, and we construct a family of Poisson structures on Y and its boundary divisors arising from Belavin-Drinfeld splittings of the double of the Lie algebra of G. In the appendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson structure on a quotient of a Poisson algebraic variety by the action of a reductive Poisson algebraic group.

math.AG↗
arXiv2016

How efficiently can one untangle a double-twist? Waving is believing!

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, geometrically defined untangling procedure, leading to new conclusions regarding the minimum possible complexity of untanglings. We animate and analyze how our untangling operates on frames in 3-space, and teach readers in a video how to wave the nullhomotopy with their hands.

math.GT↗
arXiv2014

Homotopy Groups of Free Group Character Varieties

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations. Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. This result is proven in the Appendix and may be of independent interest. We also obtain new information regarding the homotopy groups of the subspaces Y. Recent work of Biswas and Lawton determined the fundamental group of X for general G, and we describe the fundamental group of Y. Specializing to the case G = GL(n,C), we explicitly compute the homotopy groups of the smooth locus of X in a large range of dimensions, finding that they exhibit Bott Periodicity. As a further application of our methods (and in particular our general position result) we obtain new results regarding centralizers of subgroups of G and K, motivated by a question of Sikora. Additionally, we use work of Richardson to solve a conjecture of Florentino-Lawton about the singular locus of X, and we give a topological proof that for G= GL(n,C) or SL(n,C), the space X is not a rational Poincar\'e Duality Space for r>3 and n=2.

math.AT↗
arXiv2014

Covering spaces of character varieties

Let F be a finitely generated discrete group. Given a covering map H to G of Lie groups with G either compact or complex reductive, there is an induced covering map Hom(F, H) to Hom(F, G). We show that when the fundamental group of G is torsion-free and F is free, free Abelian, or the fundamental group of a closed Riemann surface M of genus g, this map induces a covering map between the corresponding moduli spaces of representations. We give conditions under which this map is actually the universal covering, leading to new information regarding fundamental groups of these moduli spaces. Let pi be the fundamental group of M. As an application, we show that for g>0, the stable moduli space Hom(pi, SU)/SU is homotopy equivalent to infinite complex projective space. In the Appendix by Ho and Liu, it is shown show that there is a bijection between the number of connected components of Hom(pi, G) and the fundamental group of [G,G] for all complex connected reductive Lie groups G.

math.AT↗
arXiv2012

A finite dimensional approach to the strong Novikov conjecture

The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using the Atiyah-Singer families index theorem, and partly by Carlsson's deformation $K$--theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the $K$--theory and cohomology of representation spaces.

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