## at.algebraic topology – basic groups of pseudo-Anosov mapping tori

I am trying to create several presentations for the basic set of surface mapping tori.

For example when $$gcd (p, q) = 1$$, then $$M =$$ the T.$$(p, q)$$ Torus knot outside in $$S ^ 3$$ is a map torus of a genus $$g = frac {pq-p-q + 1} {2}$$ Surface with a boundary component, S, which is mapped onto itself $$phi$$, a card of order $$pq$$.

The suspension flow of this mapping has closed all webs and actually results in a Seifert fiber of $$M$$. With the Seifert fiber we can express the basic group quite well in terms of the two singular fibers and the regular fiber. $$pi_1 (M) =$$. Here $$z$$ is clearly a third-party generator, but is central and includes the class of regular fibers, so somehow nice. The others $$x$$ and $$y$$ are the classes of the singular order fibers $$q$$ and $$p$$ respectively.

Since $$M$$ is a mapping torus that we also get $$pi_1 (M) = $$tb_it ^ {- 1} = phi _ * (b_i)> = pi_1 (S) rtimes _ { phi_ *} mathbb {Z}$$.

The Dehn Twist presentation by $$phi$$ is something that can be worked out, and such $$phi _ *$$ could actually be specified explicitly.

In order not to go too far, we could instead consider a surface S of the genus $$g geq 2$$ with a boundary component and a pseudo-Anosov monodromy, $$phi$$, that gives us a hyperbolic knot outside, M, in $$S ^ 3$$. We get again $$pi_1 (M) = $$tb_it ^ {- 1} = phi _ * (b_i)> = pi_1 (S) rtimes _ { phi_ *} mathbb {Z}$$.

Now we have a pseudo-Anosov suspension flow and a collection of closed, singular orbits that correspond to the singularities of the stable and unstable singular leaves. So there should be a representation of the basic group in relation to the closed singular orbits, right?

I'm assuming that I can look at Knot Info's list of hyperbolic node monodromes related to strain rotations to generate examples that I can use to calculate $$phi _ *$$ expressly. Then I could fill that out $$ta_it ^ {- 1} = phi _ * (a_i)$$ Relators. However, I would like a more general approach since I am interested in all pseudo Anosov monodromes, including narrow surfaces, not just the fiber node monodromy cards in $$S ^ 3$$. Apparently the singular courses are a candidate for a general approach.

I tried to search Thurston's Work on Surfaces (FLP) and I couldn't see such information. Has that been worked out somewhere? If so, then this is probably a reference request. If not, does anyone know of any obstacles to creating the base group with these loops?

## p adic number theory – conjugation of maximal algebraic tori

Accept $$G$$ is a connected, reductive algebraic group over a non-Archimedean local field $$F$$which is divided over a finite extent $$E / F$$,

I often see a result that says "everything is maximum $$F$$-Tori are conjugated over $$E$$", by which I understand the following: Let $$G (E)$$ denote the $$E$$-Dots of the algebraic group $$G$$;; then for each maximum $$F$$-tori $$T, T$$ of $$G$$is there $$x in G (E)$$ so that $$T (E) = xT & # 39; (E) x ^ {- 1}$$,

In addition, the definitions show that if $$T, T$$ are maximum $$F$$-tori from $$G$$then there is an isomorphism of $$T (F)$$ on to $$T & # 39; (F)$$ which is defined via $$E$$,

My question is: Can the isomorphism be assumed to be conjugation in the second statement (as in the first statement)? That means: it follows from these results that if $$T, T$$ are maximum $$F$$-tori in $$G$$then it exists $$x in G (E)$$ so that $$T (F) = xT & # 39; (F) x ^ {- 1}$$?

Any help (including proof of the first statement) is greatly appreciated!

## Abstract algebra – conjugation classes of rational tori in the symplectic group

Rational conjugation classes of Frobenius Stable Tori (in a finite Lie group) are bijected with Frobenius conjugation classes of the corresponding Weyl group$$(2n)$$ then the Frobenius action on the Weyl group is trivial and classes of rational Tori are in bijection with conjugation classes of the type C-Weyl group $$W_n$$, These conjugation classes are parameterized by bipartitions of n.

The groups of Frobenius stable points of the tori belonging to the conjugation class of the rational tori are all isomorphic. Is there a way to construct these groups from the corresponding bipartite of n? I tried Sp (4), but I can not understand much of what I have. I have also done this calculation for GL (n), for this group this is possible (and quite simple).

I hope someone can give me a hint. Thank you in advance. Hans.

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## ag.algebraic geometry – Are maximal tori conjugated locally localizable?

To let $$S$$ to be and leave a scheme $$G to S$$ to be a reductive group scheme. Then $$G$$ admits that there is a maximal torus etale-local, and two maximal tori are conjugate etale-local, according to Theorem 3.2.6 and Corollary 3.2.7 in [B. Conrad, Reductuve group schemes] http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

As stated in the book under Proposition 3.1.9 and at the beginning of Chapter 4, the presence of maximum Tori Zariski locally can be achieved (cf. [SGA3, XIV, 3.20]). My question is the following

To let $$T$$ and $$T #$$ be maximum tori $$G$$ over defined $$S$$are they conjugated Zariski-local?

In my case, $$T$$ is split $$S$$, and $$S$$ is affine with $$Pic (S) = 0$$,

## ag.algebraic geometry – Tori classes in the Grothendieck variety ring

Inspired by this question, I wondered if it is possible to write down the class of a normative torus $$K_0 ( operatorname {var} _k)$$ only use $$mathbb {L}$$ and classes of étale algebras. I know it's possible for quasi-split Tori $$R_ {K / k} ( mathbb {G} _m)$$by an inclusion-exclusion principle (see this document by Rökaeus), hence for the Formi $$R_ {K / k} ( mathbb {G} _m) / mathbb {G} _m$$and the question I ask relates to the duals of it.

I can not say anything in the case $$[K:k]= 4$$,

Note: If a torus can be written as a polynomial in $$mathbb {L}$$ with coefficients classes of the étalen algebras, then passing on $$l$$-adic cohomology become the coefficients $$lambda$$– Forces of the character grid of the torus.

A proof that Norm-1-Tori in general can not only be written with $$mathbb {L}$$ Even more interesting would be the teaching of étaler algebras.

## algebraic groups – maximum tori of a symmetric subgroup

Accept $$G$$ is a complex connected reductive algebraic group, $$K$$ is a symmetric subgroup of $$G$$ (ie the fixed points of an involution $$theta$$ from $$G$$), and $$T$$ is a $$theta$$– stable maximum torus in $$G$$, Which assumptions do we need? $$G$$, $$K$$, and $$T$$ in order to $$S = K cap T$$ is connected and a maximum torus of $$K$$?