arXiv research

Proves Burau representation is faithful for 4-strand braids.

Study on faithfulness of Burau representation for Artin-Tits groups.

The Burau representation of 3-strand braid group modulo p is determined and shown to be faithful for small p.

We prove asymptotic faithfulness for the quantum $\mathrm{Sp}(4)$ mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the $A_n$ family. The methods used are generalized from the proof of asymptotic faithfulness for skein $SU(2)_k$ mapping class group represent…

The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$,…

We analyze two braid group representations and their reductions modulo p.

The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton imp…

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a question of W. Goldman. We also prove the existence of faithful representations into PU…

Method produces faithful representations of Garside groups and torus knot groups.

The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is 1 and 2. In this article we show that the Lawrence representation is faithful for $m \geq 2$.

The paper explores representations of graph manifolds to Seifert motion groups.

We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.

Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

Discrete PU(1,1) representations of hyperelliptic groups are proven.

Link between braid groups and q-deformed rationals solves a classification problem.

We generalize the asymptotic faithfulness of the skein quantum $SU(2)$ representations of mapping class groups of orientable closed surfaces to skein $SU(3)$. Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier …

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful repres…

Improved lower bounds for faithful linear representations of mapping class groups.

The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.

We prove that the hyperelliptic mapping class group of a nonorientable surface of genus $g\geq 4$ has a faithful linear representation of dimension $g^2-1$ over $\mathbb{R}$.

Counterexamples show Salter's question on Burau image is negative for n=4.

Researchers compute TQFT representation for sphere with 4 punctures.

The Burau representation of braid group B4 is shown to be faithful almost everywhere.

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the c…

The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.

\begin{abstract} The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;\mathbb{Z})$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$--punctured disc $D_n$. It is known that the Burau representation is faithful for $n\le 3$ and…

We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and…

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a fa…

In a previous work [11], the author considered a representation of the braid group ρ: B_n\to GL_m(\Bbb Z[q^{\pm 1},t^{\pm 1}]) (m=n(n-1)/2), and proved it to be faithful for n=4. Bigelow [3] then proved the same representation to be faithful for all n by a beautiful topological argument. The present paper gives a diffe…

In this paper we give a criterion for pairs of isometries of a nonpositively curved metric space to generate a free group. This criterion holds only in singular spaces, for example in Euclidean buildings. The original motivation for our criterion was to prove that the four dimensional Burau representation is faithful. …

In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…

This study finds a special class of representations that dominate others in a complex hyperbolic group.

Ideal attribution mechanisms track model interactions for faithful watermarks.

We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.

We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…

In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely r…

This paper aims to generalize Artin's ideas to establish an one-to-one correspondence between the orbit braid group $B^{orb}_n(\mathbb{C},\mathbb{Z}_p)$ and a quotient of a group formed by some particular homeomorphisms of a punctured plane. First, we find a faithful representation of $B^{orb}_n(\mathbb{C},\mathbb{Z}_p…

Complex hyperbolic triangle groups are discrete when certain conditions are met.

We categorify the coefficients of the Burau representation matrix using elementary geometrical methods. We show the faithfulness of this categorification in the sense that it detects the trivial braid.

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie…

We give an exposition of the work of Bigelow and Krammer who proved that the Artin braid groups are linear.

COCA refactors training data to identify and erase unsafe concepts in LLMs.

Geodesic patterns, shears, and Anosov representations of the modular group.

In this paper, we give a complete criterion for a discrete faithful representation $ρ:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/ρ(F_n)$ regarding the primitive stability of $ρ$.

Any nontrivial homomorphism from the mapping class group of an orientable surface of genus $g\geq 3$ to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful linear representation in dimensions less than or equal to $3g-3$.