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To expand on my comment above : as noted by john mangual $SU(1,1)$ is isomorphic to $SL(2, \mathbb R)$. If $h$ is an Hermitian form defined over a ring of integers $\mathcal O_K$, where $K$ is imagina …

It is a fact that if $\Lambda$ is a nonelementary subgroup of ${\rm PSL_2}(\mathbb{C})$ which contains an hyperbolic transformation and moreover ${\rm tr}(g)\in\mathbb{R}/\pm 1$ for all $g\in\Lambda$ …

I'll try to flesh out the first comment to give an answer: for all $n\ge 4$ there is a representation $\rho:A_{n+1}\to \rm SL_n(\bf Z)$ whose image normally generates $\rm SL_n(\bf Z)$ (because it inj …

I guess the folllowing is a short explicit proof of Tits' result for this very special case, assuming CSP for $\rm{SL}_n(\bf{Z})$ (so this should be seen as a lenghty comment on the above answer). Th …

Such a group has been found by N. Dunfield, see the appendix to the paper Steffen Kionke, Jean Raimbault, Nathan Dunfield, On geometric aspects of diffuse groups, Documenta Mathematica, Vol. 21 (2016 …

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies $(\dag …

There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and sta …

I am interested in the following problem : given a finite field $F$ and two unipotent elements $g_1,g_2\in\mathrm{SL}_2(F)$ which do not commute, what can we say about the subgroup they generate? More …

If $G$ is a group let $I(G)$ be the number of involutions (elements of order 2) in $G$. My question is then easily stated: does there exists a constant $C > 1$ such that for every $n \ge 1$ and every …

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other "repelling". …

This is not a complete answer but it's too long to fit in a comment, so here it goes. In case $g \in \mathrm{SL}_2(\mathcal O_k)$ is semisimple there are two possibilities : As you observed, if $g$ …

This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly dif …

This is not the case, in fact in both cases the limit is 1/2. This is because with probability going to 1 as $n \to +\infty$ a uniformly random morphism $\mathbb F_2 \times \mathbb F_2 \to \mathrm{Sym …

This is true in general, by an argument similar to the one you used for the coprime case. The subgroup $\Gamma$ generated by $\Gamma(N_1) \cup \Gamma(N_2)$ contains the matrices $$ a = \begin{pmatrix …

Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition? On …