Third homology of general linear groups

Homology of Groups and Third Busy Beaver Function

In 2007, Nabutovsky and Weinberger provided a solution to a longstanding problem: to find naturally defined functions that grow faster than any function with Turing degree of unsolvability 0′. They considered the functions bk such that, for a natural integer N , bk(N) is the rank of the kth homology group Hk(G) of maximum finite rank, among the finitely presented groups G with presentation leng...

characterization of projective general linear groups

let $g$ be a finite group and $pi_{e}(g)$ be the set of element orders of $g $. let $k in pi_{e}(g)$ and $s_{k}$ be the number of elements of order $k $ in $g$. set nse($g$):=${ s_{k} | k in pi_{e}(g)}$. in this paper, it is proved if $|g|=|$ pgl$_{2}(q)|$, where $q$ is odd prime power and nse$(g)= $nse$($pgl$_{2}(q))$, then $g cong $pgl$_

Rigidity of the Low Dimensional Homology of Linear Groups

Homology of Linear Groups via Cycles in Bg×x

Homology theories for algebraic varieties are often constructed using simplicial sets of algebraic cycles. For example, Bloch’s higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X,G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG×X, an...

The Homology of Special Linear Groups over Polynomial Rings

We study the homology of SLn(F [t, t ]) by examining the action of the group on a suitable simplicial complex. The E–term of the resulting spectral sequence is computed and the differential, d, is calculated in some special cases to yield information about the low-dimensional homology groups of SLn(F [t, t ]). In particular, we show that if F is an infinite field, then H2(SLn(F [t, t ]), Z) = K...