Write \(M\) for the maximum in the problem. The pair \[ A=\begin{pmatrix}69&82\\70&51\end{pmatrix},\qquad B=\begin{pmatrix}41&48\\47&92\end{pmatrix} \quad(\bmod 101) \] generates all 1,030,200 elements of \(\mathrm{SL}_2(\mathbf F_{101})\), and its undirected Cayley graph has girth exactly 17. This proves \(M\geq17\).

Every graph under consideration is 4-regular on \[ 101(101^2-1)=1{,}030{,}200 \] vertices. A 4-regular graph of girth at least 25 would contain a tree ball with \[ 1+4\sum_{i=0}^{11}3^i=1{,}062{,}881 \] distinct vertices. This exceeds the group order, so \(M\leq24\). Together, \[ 17\leq M\leq24. \] The upper bound ranges over every eligible pair. The lower-bound computation certifies one pair and does not enumerate simultaneous-conjugacy classes of all pairs.

Suppose a 4-regular graph has girth at least 25. The nonbacktracking paths of length at most 12 starting at a fixed vertex have distinct endpoints. Otherwise two such paths would give a cycle of length at most 24. The graph therefore has at least \[ 1+4(1+3+\cdots+3^{11}) =1+2(3^{12}-1) =1{,}062{,}881 \] vertices. The Cayley graphs in this problem have 1,030,200 vertices, so their girth is at most 24.

Use lowercase letters for inverses. Exact multiplication modulo 101 gives \[ A^4BA B^{-1}A^{-2}B^{-2}A^{-1}B^5=I. \] In letter form, one such relation is `AAAABAbaabbaBBBBB`, of length 17.

For the lower bound, enumerate every freely reduced word of length at most eight. Their matrix values are all distinct. The layer sizes are \[ 1,4,12,36,108,324,972,2916,8748, \] whose sum is 13,121, exactly the radius-eight ball in the 4-regular tree. Any relation of length at most 16 can be split into two words of length at most eight with the same matrix value. The injective word ball rules this out. An edge between two vertices in the outer layer supplies the displayed length-17 relation, so the girth is 17.

A separate full BFS reaches 1,030,200 determinant-one matrices. Since \(|\mathrm{SL}_2(\mathbf F_{101})|=101(101^2-1)=1{,}030{,}200\), the matrices generate the stated group. Their inverses and one another are pairwise distinct, so the Cayley graph has degree four.

Lubotzky, Phillips, and Sarnak construct arithmetic Cayley graphs with asymptotically large girth. Their groups are \(\mathrm{PSL}_2\) or \(\mathrm{PGL}_2\) for selected arithmetic parameters, and their result does not enumerate two-generator pairs in \(\mathrm{SL}_2(\mathbf F_{101})\).

Gamburd, Hoory, Shahshahani, Shalev, and Virag prove asymptotic lower bounds for the girth of random Cayley graphs of simple algebraic groups. Their \(\mathrm{SL}_2\) result concerns probability as the field grows. Bourgain and Gamburd use logarithmic girth as an input to uniform expansion results. Neither paper reports an exact maximum for this finite group.

The targeted search covered the exact group name, the prime 101, two-generator Cayley graphs, random Cayley girth, LPS graphs, and computational girth tables. No primary source giving the value of \(M\) or improving the finite ceiling 24 was located. This is a search report dated 2026-07-25, so novelty remains unverified.

Matrices are four-tuples \((a,b,c,d)\) with exact arithmetic modulo 101. The word-ball pass omits only immediate inverse steps. It checks the full free-tree layer vector through radius eight and rejects any repeated endpoint. It then scans the outer sphere for internal edges and verifies the lexicographically first resulting length-17 relation.

The generation pass encodes a matrix as \[ a+101\bigl(b+101(c+101d)\bigr) \] and uses a dense byte array for visited states. It reaches the full group order and checks the determinant of every reached matrix. The radius-eight endpoint table has SHA-256 digest `5a74202c3e18b637c5dbb63e89080cc4a4773101f13e1ecdf4fa0b64efba8bcb`. The portable little-endian BFS queue digest is `26d6373b5b6475e5d99ccf47a500d2b33ca00dd23793d80fed6893af0ba559a2`.