18.90 (2014)

Prove that the group $Y(m, n) = \langle a_1, a_2, \dots, a_m \mid a_k^n = 1 \, (1 \leqslant k \leqslant m), (a_k^i a_l^i)^2 = e \, (1 \leqslant k < l \leqslant m, 1 \leqslant i \leqslant \frac{n}{2}) \rangle$ is finite for every pair $(m, n)$.

Some known cases are (where $C_n$ denotes a cyclic group of order $n$): $Y(m, 2) = C_2^m$; $Y(m, 3) = A_{m+2}$ (presentation by Carmichael); $Y(m, 4)$ has order $2^{m(m+3)/2}$, nilpotency class 3 and exponent 4; $Y(2, n)$ is the natural extension of the augmentation ideal $\omega(C_n)$ of $\text{GF}(2)[C_n]$ by $C_n$ (presentation by Coxeter); $Y(3, n) \cong \text{SL}(2, \omega(C_n))$.

These groups have connections with classical groups in characteristic 2. When $n$ is odd, the group $Y(m, n)$ has the following presentation, where $\tau_{ij}$ are transpositions:
$$y(m, n) = \langle a, S_m \mid a^n = e, [\tau_{12}^{a^i}, \tau_{12}] = e \, (1 \leqslant i \leqslant n/2), \tau_{12}^{1+a+\dots+a^{n-1}} = e, \tau_{i, i+1} a \tau_{i, i+1} = a^{-1} \, (2 \leqslant i \leqslant m-1) \rangle,$$ and it is known that, for example, $y(3, 5) \cong \text{SL}(2, 16) \cong \Omega^-(4, 4)$, $y(4, 5) \cong \text{Sp}(2, 16) \cong \Omega(5, 4)$, $y(5, 5) \cong \text{SU}(4, 16) \cong \Omega^-(6, 4)$, $y(6, 5) \cong 4^6 \Omega^-(6, 4)$, $y(3, 7) \cong \text{SL}(2, 8)^2 \cong \Omega^+(4, 8)$, $y(4, 7) \cong \text{Sp}(4, 8) \cong \Omega(5, 8)$. Extensive computations by Felsch, Neubüser and O'Brien confirm this trend. Different types reveal Bott periodicity and a connection with Clifford algebras.