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Results tagged with gr.group-theory
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user 16143
Questions about the branch of algebra that deals with groups.
23
votes
What invariants of a matrix or representation can be used to find its GL(n,Z)-conjugacy class?
In each conjugacy class you can always find a representative which is block upper triangular, and the diagonal blocks have irreducible characteristic polynomials. This gives a partial reduction to the …
- 3,201
21
votes
4
answers
3k
views
Computing the Zariski closure of a subgroup of SL(n,Z)
Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is …
- 3,201
13
votes
Accepted
Lattices in $SL(n,\mathbb R)$
The answer is yes. It is theorem [2.13] of the following paper of Prasad and Raghunathan:
Prasad, Gopal; Raghunathan, M. S. Cartan subgroups and lattices in semi-simple groups. Ann. of Math. (2) 96 ( …
- 3,201
9
votes
Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z...
By the Margulis superrigidity theorem, any homomorphism from $SL(n,\mathbb{Z})$ to $SL(n-1,\mathbb{R})$ with infinite image has to extend to a homomorphism from $SL(n,\mathbb{R})$ to $SL(n-1,\mathbb{R …
- 3,201
8
votes
Ergodic action of a subgroup
You can take $G$ to be any simple Lie group with finite center, and $H$ to be any non-compact Lie subgroup. Then, if the action of $G$ is ergodic, so is the action of $H$. This statement is called the …
- 3,201