BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Johann Thiel (New York City College of Technology (CUNY)) DTSTART:20220524T193000Z DTEND:20220524T195500Z DTSTAMP:20260423T011438Z UID:CANT2022/11 DESCRIPTION:Title: Solving the membership problem for certain subgroups of $SL_2(\\mathbb{Z })$\nby Johann Thiel (New York City College of Technology (CUNY)) as p art of Combinatorial and additive number theory (CANT 2022)\n\n\nAbstract\ nFor positive integers $u$ and $v$\, let $L_u=\\begin{bmatrix} 1 & 0 \\\\ u & 1 \\end{bmatrix}$ and $R_v=\\begin{bmatrix} 1 & v \\\\ 0 & 1 \\end{bma trix}$. Let $G_{u\,v}$ be the group generated by $L_u$ and $R_v$. The memb ership problem for $G_{u\,v}$ asks the following question: Given a 2-by-2 matrix $M=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}$\, is there a rela tively straightforward method for determining if $M$ is a member of $G_{u\ ,v}$? In the case where $u=2$ and $v=2$\, Sanov was able to show that simp ly checking some divisibility conditions for $a$\, $b$\, $c$ and $d$ is en ough to make this determination. We answered this question in the case whe re $u\,v\\geq 3$ by finding a characterization of matrices $M$ in $G_{u\,v }$ in terms of the short continued fraction representation of $\\frac{b}{d }$\, extending some results of Esbelin and Gutan. By modifying our previou s work\, we are able to further extend our previous result to the more dif ficult case where $u\,v\\geq 2$ with $uv\\neq 4$.\n\nThis is joint work wi th Sandie Han\, Ariane M. Masuda\, and Satyanand Singh.\n LOCATION:https://researchseminars.org/talk/CANT2022/11/ END:VEVENT END:VCALENDAR