## gr.group theory – Braid groups and Kazhdan’s property (T)

In Nica’s dissertation Group actions on median spaces, we can read the following assertion:

Braid groups do not contain infinite subgroups satisfying Kazhdan’s property (T).

This is used in order to motivate the question (which is still open up to my knowledge) of whether braid groups satisfy the Haagerup property. Does anyone have a reference and/or a quick justification of the above claim?

## gt.geometric topology – How to get the braid word of the tubes in a reducible braid?

If a braid is reducible, it means that the strands can be separated into groups of strands (less groups than number of strands), such that each group is running inside one tube. The tubes themselves can form a braid.

I am aware of algorithms that determine whether a braid is reducible, implemented for example in flipper. Is it also possible to determine a braid word for the braid that is formed by the tubes? (Perhaps one can get it directly from another output, in that case my understanding is not developed enough to know how.)

Is the special case of determining whether the constituent braid formed of the tubes is trivial implemented somewhere?

## Word problem in the braid group

The geodesic length of elements can be defined as the length of a minimal path from 1 to w in the Cayley graph of G. This length is dependent on the particular generating set X.
The Word Problem is decidable in G if and only if the geodesic length function is computable.

Sufficiency is obvious, but why is the necessity needed? Further, “computable” I interpret as the calculation terminates in a finite number of steps, with the correct answer, is this correct, or is it more nuanced?

The braid group has a quadratic Dehn function, i.e., there exist constants $$C_n$$, $$C’_n$$ such that, if w is an n-strand braid word of length $$l$$ that represents the unit braid, then the number of braid relations needed to transform $$w$$ into the empty word is at most $$C_n l^{2}$$ and, on the other hand, there exists for each $$l$$ at least one length $$l$$ word $$w$$ such that the minimal number of such braid relations is at least $$C’_n l^2$$. (This paragraph is copied from Combinatorial Distance between Braid Words, P. Dehornoy, which mentions it is a well-known result)

Because the first result is if and only if, this, together with the braid group having a quadratic Dehn function, (presumably) means that word problem in the braid group is decidable. Does this mean that the Dehn function can be used to calculate the geodesic length in the braid group?

## homotopy theory – Integral homology of braid groups as a ring

Let $$Br_k$$ denote the braid group on $$k$$ strands. In Corollary A.4 of “Homology of Iterated Loop Spaces” (Page 348), Cohen-Lada-May compute $$H_i(Br_k;mathbb Z)$$ as an abelian group for each $$i$$ and $$k$$. There is a ring structure on $$bigoplus_{k,i} H_i(Br_k;mathbb Z)$$ induced by the maps $$Br_k times Br_j to Br_{k+j}$$. Is the ring structure on $$bigoplus_{k,i} H_i(Br_k;mathbb Z)$$ known? Cohen-Lada-May compute it with field coefficients. Equivalently, is the ring structure on $$H_*(Omega^2 S^2;mathbb Z)$$ known?

## cryptography – A hash function on braid groups

I am reading group theoretic cryptography. I wanted an example of a hash function from the braid group $$B_n$$ to a fixed size string of $$0$$s and $$1$$s, say $${0,1}^k$$. I am a group theorist and haven’t really dealt with such functions in depth, but I really want to know a nice application on $$B_n$$. Since every braid has a unique Garside’s normal form (one can read more here), maybe we could construct using it?

## rt.representation theory – Conceptual Proof of Braid Group Actions on Quantum Groups

Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $$U_q(mathfrak{g})$$ is the proof which is least conceptual.

The original paper contains a case by case check for $$m=2,3,4,6$$ as far as I understand.

If it is not a coincidence, then it needs an explanation. My question is

Are there any conceptual proof up to now is known?

At least not one by one.

• For example, can we define quantum groups for the type $$I_n$$? (The naive version is to work in the field of Puiseux series.)

• Perhaps we have very limited ‘conceptual’ understanding on quantum groups up to now. There is a lot on the negative part $$mathfrak{f}$$ (geomatric realization, categorification, etc.). But the braid group action is on the complete quantum group. The classic one is the Hall algebra. For this are there any interpolation of the braid group actions?

• The only other result I know is the work of Nakajima which showed that the whole algebra can be realized by the equivariant K-theory of Nakajima varieties. (any result about how this action reflecting the geometry side?)

• For example, the simplest case, in the springer realization of $$U_q(mathfrak{sl})_3$$, does the braid group action have any geometric meaning?

## gr.group theory – Do Ternary Braid Groups Appear in Algebraic Topology?

Let $$TB_ {n}$$ to be the group defined by the presentation with generators $$t_ {1}, …, t_ {n-2}$$ and relationships $$t_ {i} t_ {i + 1} t_ {i + 2} t_ {i} = t_ {i + 2} t_ {i} t_ {i + 1} t_ {i + 2}$$
and $$t_ {i} t_ {j} = t_ {j} t_ {i}$$ does not matter when $$| i-j |> 2$$. We will call $$TB_ {n}$$ the ternary braid group. The group $$TB_ {n}$$ has many of the characteristics of the braid group $$B_ {n}$$. For example, there is an injective group homomorphism
$$phi: B_ {n} rightarrow TB_ {2n}$$ Defined by $$phi ( sigma_ {i}) = t_ {2i-1} t_ {2i}$$. Then group $$TB_ {n}$$ can also be endowed with a ternary self-distribution operation similar to the displaced conjugation. The group $$TB_ {n}$$ even acts on baskets and ternary dilemmas.

In addition to $$TB_ {n}$$There are also several topological ways to generalize the group of braids to various groups. For example, groups of braids appear as groups of mapping classes and basic groups of some topological spaces. Braids of larger dimensions can also be envisaged. is $$TB_ {n}$$ isomorphic to a group of braids of greater or general size? Is there a group of more dimensional or generalized braids? $$G$$ with a homomorphism $$phi: TB_ {n} rightarrow G$$ of a small nucleus or a homomorphism $$theta: G rightarrow TB_ {n}$$ with small nucleus? Informally, the group $$TB_ {n}$$ almost arise in the algebraic topology?

## group theory – does the Hurwitz action of the braid group on rank integrations in ranks tend to increase critical points?

An algebraic structure $$(X, *)$$ is said to be self-distributing if it satisfies the identity $$x * (y * z) = (x * y) * (x * z)$$.

Assume that $$X$$ is a self-distributing algebra. Then the monoid with positive braid $$B_ {n} ^ {+}$$ acts on $$X ^ {n}$$ leaving
$$(x_ {1}, …, x_ {n}) cdot sigma_ {i} = (x_ {1}, …, x_ {i-1}, x_ {i} * x_ {i + 1}, x_ {i}, x_ {i + 2}, …, x_ {n})$$ and this action is called Hurwitz action.

Recall that an algebraic structure $$(X, *)$$ is cancelable on the left if $$x * y = x * z$$ involved $$y = z$$. Yes $$X$$ is cancelable on the left, then this action extends to a partial action of the group of braids $$B_ {n}$$ sure $$X ^ {n}$$ by defining
$$(x_ {1}, …, x_ {n}) cdot bc ^ {- 1} = (y_ {1}, …, y_ {n})$$ precisely when
$$(x_ {1}, …, x_ {n}) cdot b = (y_ {1}, …, y_ {n}) cdot c$$, and this partial action still calls Hurwitz 's action.

Let $$mathcal {E} _ { lambda} ^ {+}$$ be all of all elementary traffic jams $$j: V _ { lambda} rightarrow V _ { lambda}$$. then $$mathcal {E} _ { lambda} ^ {+}$$ can be endowed with a self-distributing operation $$*$$ defined leaving
$$j * k = bigcup _ { alpha < lambda} j (k | _ {V _ { alpha}}$$. L & # 39; operation $$*$$ however, it is neither associative nor commutative. Cardinal $$lambda$$ is the limit of many very great cardinals. For more information on the algebraic structure $$( mathcal {E} _ { lambda} ^ {+}, *)$$please refer to Chapter 11 of the Handbook of Set Theory. For more information on the two braids, Hurwitz's action and basic operations, refer to Chapters 1, 3 and 12 of Patrich Dehornoy's book Braids and Self-Distributivity.

Assume that $$j_ {1}, …, j_ {n} in mathcal {E} _ { lambda} ^ {+}$$ and $$gamma < lambda$$. Remind that $$mathrm {crit} (j)$$ denotes the smallest ordinal $$alpha$$ or $$j ( alpha)> alpha$$. So is there only finely braids $$b in B_ {n}$$ as if $$(j_ {1}, …, j_ {n}) cdot b = (k_ {1}, …, k_ {n})$$then $$mathrm {crit} (k_ {1}) < gamma, …, matthm {crit} (k_ {n}) < gamma$$? If we limit ourselves to positive braids, then the answer is a yes by my answer here.