Lyn Assy - Some examples on virtual representations

Abstract

Talk

We start with the motivation. Lyn describes the similarities between the Automorphism group of Fn and the Mapping class group.

Definition

A group is linear if there is a finite-dimensional faithful representation.

Example

• $Aut(F_2)$ is linear
• $Aut(F_3)$ is not linear

Let $\rho :X\to R_3$ be a regular cover.

We describe two Goals:

1. Find a representation of $Aut(F_3)\to GL(H_1(X,\mathbb{C}))$
2. Find two elements in $Aut(F_3))$ that act trivially on $H_1(R_3)$ but lift to elements that act nontrivially on $H_1(X)$

We define what a normal cover is and paint a picture of a cover which is not normal (something where the deck group does not act transitively on the fibres)

Theorem (Garschütz)

As a $G$-representation, we have $H_1(X,\mathbb{C})=\mathbb{C}[G{]}^{n-1}\oplus \mathbb{C}$

Theorem

A theorem by Maske tells us, that the regular $G$-representation of $V$ carries all irreducible representations.

Claim 1: As a $G$-representation. $H_1(X)$ decomposes as $T\oplus K$.

We now study the example given of $R_3$ (consisting of curves $a,b,c$) with a two-fold cover (cut open along $c$). We get the following homological maps:

• $a_1+a_2\stackrel{C_2}{\mapsto }{a}_2+a_1\stackrel{p_{\ast }}{\mapsto }2a$
• $b_1+b_2\stackrel{C_2}{\mapsto }{b}_2+b_1\stackrel{p_{\ast }}{\mapsto }2b$

This is an example of a subspace of the virtual homology, on which $C_2$ acts trivially.

The following subspace is the kernel of $ker(H_1(X)\to H_1(R_3))$. Notice that the Deck transformation acts non-trivially on the basis $a_1-a_2$ and $b_1-b_2$.

• $a_1-a_2\stackrel{C_2}{\mapsto }{a}_2-a_1\stackrel{p_{\ast }}{\mapsto }0$
• $b_1-b_2\stackrel{C_2}{\mapsto }{b}_2-b_1\stackrel{p_{\ast }}{\mapsto }0$

Together with the curve $c_1+c_2$ we get a basis, s.t. the Deck group acts as the matrix $\left(\begin{array}{ccccc}1& & & & \\ & 1& & & \\ & & 1& & \\ & & & -1& \\ & & & & -1\end{array}\right)$ on the homology of the cover.

Claim 2: Let $\Gamma$ be a nice subgroup of the automorphism group $Aut(F_3)$. Precisely, it is the group of Automorphisms of $F_3$ (understood as a map of Homotopy e,quivalence) that lifts to the cover. (The exact condition being $f_{\ast }(p_{\ast }({\pi }_1(X,\ast )))=p_{\ast }{\pi }_1(X,\ast )$, where $X$ is the cover)

Definition

A characteristic cover is a cover, where every Homotopy equivalence/Homeomorphism lifts to a homotopy equivalence/homeomorphism on the cover.

Claim 3: There is a $f\in ker(\rho )$ s.t. $\tilde{f}\notin ker(\tilde{\rho })$. ($f$ acts trivially on $R_3$ but not on the cover $X$.) Take $f:(R_3,v)\to (R_3,v),a\mapsto a[b,c],b\mapsto b,c\mapsto c$.