A new mathematical preprint gives a clearer map of the homology of the groups SL2(Z[1/n]), showing that their first homology has a four-way classification while higher degrees can be bounded through a construction involving Bruhat–Tits trees.
The work concerns theoretical objects rather than an empirical sample. Its central question is how to calculate or constrain the integral homology groups H_k(SL2(Z[1/n]), Z) as n varies, using group actions on products of these trees.
A tree-based route through the calculation
The main tool is a spectral sequence—a staged algebraic calculation—that converges to the homology of SL2(Z[1/n]). Its first stage is built from the homology of congruence subgroups written Γ0(p_i1⋯p_is), with multiplicities determined by the s-element subsets of the prime factors of n.
The text reduces the problem to n written as a product of distinct primes and says it is enough to study square-free n. The resulting framework also compares SL2 with PSL2 and uses the associated tree quotients, stabilizers and spectral-sequence data.
Calculations for Γ0(n) supply part of the input. The dissertation gives a case-based description: its degree-zero homology is Z; degree one has the form Z^{r(n)}⊕G; even degrees above one have (Z/2)^{r(n)}; and odd degrees above one have G, with G determined by b(n) and c(n).
The first homology is completely pinned down
One of the sharpest results is the classification of H1(SL2(Z[1/n])). For every positive integer n, it is zero when both 2 and 3 divide n; it is Z/3 when 2 divides n but 3 does not; Z/4 when 3 divides n but 2 does not; and Z/12 when neither 2 nor 3 divides n.
The same analysis says that the rational first homology vanishes for all integers n. In ordinary terms, after rational coefficients are used, no first-degree free component remains; for non-zero n, the integral first homology is therefore torsion.
Exact answers in special cases, limits in general
Finite generation holds in every degree for both SL2(Z[1/n]) and PSL2(Z[1/n]). When n is a prime p, the rank pattern is especially spare: rank 1 in degree zero, r(p) in degree two, and zero in every other degree.
For n with k prime factors, the rational homology vanishes above degree k+1. In degrees 2 through k+1, the ranks are bounded above by a formula built from r-values attached to products of (s−1) prime factors, with a factor 2^(k−s+1). The bound is not claimed to be sharp in general.
The analysis improves the degree-two information when a prime divisor p has the smallest r(p) among the prime divisors of n: the rank of H2 lies between 1 and r(p). If n is divisible by 2, 3, 5, 7 or 13, that rank is exactly 1.
For square-free n, ordering the prime factors by their r-values gives a stronger upper bound for the rank of H3. The result remains a bound, not a general equality, and applies within the stated square-free setting.
A more explicit answer is available when n=pq for distinct primes and one of them is 2, 3, 5, 7 or 13. The ranks are then 1 in degrees zero and two, r(pq)−2r(p)−2r(q)+1 in degree three, and zero otherwise.
A framework, not a complete higher-degree catalogue
The result is a framework for controlling these groups, not a complete higher-degree catalogue for arbitrary n. The general statement supplies vanishing results and upper bounds, while exact rank formulas are reserved for the prime case and the specified two-prime setting.
Several conclusions depend on conditions such as square-freeness, an ordering by r, or the presence of one of five specified prime divisors. Outside those settings, the paper does not claim that its upper bounds are equalities.
Paper data and sources
Original title: The integral homology of $\text{SL}_2(\mathbb{Z}[1/n])$
Authors: Isadora Vanzella Picinini, Behrooz Mirzaii
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text