All Questions
Tagged with algebraic-k-theory gr.group-theory
7 questions
19
votes
7
answers
3k
views
Universal cover of SL2(R) admits no central extensions?
Is it true that the universal cover of $\mathrm{SL}_2(\mathbb{R})$ has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective ...
- 43.2k
26
votes
1
answer
1k
views
Group with finite outer automorphism group and large center
Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...
- 63.9k
18
votes
4
answers
2k
views
For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ ...
- 6,614
4
votes
2
answers
578
views
Suslin's Stability Theorem for Chevalley Groups
I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...
- 6,224
20
votes
1
answer
1k
views
Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?
Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
- 7,893
16
votes
3
answers
797
views
For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $...
- 563
9
votes
0
answers
426
views
Is the class of commutative generalized Euclidean rings stable under quotient and localization?
Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
- 7,893