Uniform torsion bounds for homology of arithmetic and S-arithmetic lattices and Soulé's conjecture on torsion in the K-theory of rings of integers

Emery (Math. Z. 2014) proved that, under a rational-homology vanishing hypothesis, the logarithm of the torsion subgroup of the integral homology of an irreducible nonuniform arithmetic lattice is bounded linearly by the covolume, and deduced from results of Calegari and Venkatesh discriminant-polynomial upper bounds for $K_2$ of the rings of integers (and rings of S-integers) of totally imaginary number fields. The questions that the paper explicitly records as open are: removing the rational-vanishing hypothesis from the torsion-homology bound; establishing a Gelander-type volume-linear cell-complex bound for S-arithmetic groups, which is the missing geometric input for a uniform treatment of small degree; and, as the conjecturally optimal target on the K-theory side, Soulé's conjecture that $\log |K_m(\mathcal{O}_F)_{\mathrm{tors}}|$ grows at most logarithmically in the absolute value of the discriminant $D_F$ for number fields $F$ of fixed degree. The paper additionally acknowledges, though it does not pose them as explicit open questions, that its constants $C(d)$ and $C(d,N)$ are not effective and that its discriminant exponents may be far from sharp.

Background

A number field $F$ has a ring of integers $\mathcal{O}_F$ and an integer invariant $D_F$, the discriminant, which measures the arithmetic complexity of $F$; for example the class number $h_F$ satisfies the classical bound $h_F \le C(d)|D_F|^{1/2}(\log |D_F|)^{d-1}$ for fields of degree $d$. On the geometric side, let $G$ be a connected semisimple real algebraic group whose group of real points $G(\mathbb{R})$ has trivial center and no compact factor, and let $X = G(\mathbb{R})/K$ be the associated symmetric space of noncompact type, i.e. the quotient by a maximal compact subgroup $K$. A lattice $\Gamma \subset G(\mathbb{R})$ is a discrete subgroup of finite covolume; it is arithmetic if it is commensurable with the integer points of a $\mathbb{Q}$-algebraic group, irreducible if it is dense in each nontrivial direct factor of $G(\mathbb{R})$, and nonuniform if the quotient $\Gamma \backslash X$, a locally symmetric orbifold, is noncompact (has cusps).

Beyond the rational homology (Betti numbers), which a classical theorem of Gromov bounds linearly in the volume and Samet extended to orbifolds that may come from lattices with torsion, there is the torsion subgroup $H_j(\Gamma)_{\mathrm{tors}}$ of the integral homology, of great arithmetic interest. Gelander (Duke Math. J. 2004) proved that every noncompact arithmetic manifold is homotopy equivalent to a simplicial complex whose number of cells in each dimension is at most a constant times the volume; by a lemma of Gabber bounding the torsion of a cokernel in terms of the sizes of the boundary-matrix entries, this yields $\log |H_j(\Gamma)_{\mathrm{tors}}| \le \alpha \cdot \mathrm{vol}(\Gamma \backslash X)$ for torsion-free nonuniform arithmetic lattices. Emery (Math. Z. 2014) extended this to irreducible nonuniform arithmetic lattices that may contain torsion, under a hypothesis: he required $H_q(\Gamma, \mathbb{Q}) = 0$ for $q = 1, \ldots, j$ for all irreducible lattices $\Gamma \subset G(\mathbb{R})$. The proof passes to a torsion-free normal subgroup of uniformly bounded index with finite quotient group $\mathcal{G}$ and analyzes the Lyndon–Hochschild–Serre spectral sequence, a standard tool expressing the homology of $\Gamma$ through the homology of $\mathcal{G}$ with coefficients in the homology of the subgroup; the obstruction to dropping the hypothesis is that one would need to bound the torsion in $H_p(\mathcal{G}, A_{\mathrm{free}})$, where $A_{\mathrm{free}}$ is the free part of a homology group viewed as a module over the finite group $\mathcal{G}$, and indecomposable integral representations of a finite group can be arbitrarily large, so no uniform pointwise control of the relevant boundary maps is available.

A second gap concerns $S$-arithmetic groups: for a finite set $S$ of places of $F$ containing the archimedean ones, the ring of $S$-integers $\mathcal{O}_F(S) = \mathcal{O}_F[S^{-1}]$ gives rise to groups such as $\mathrm{SL}_2(\mathcal{O}_F(S))$, which act on the product of the symmetric space with the Bruhat–Tits trees of $\mathrm{SL}_2$ over the nonarchimedean places in $S$ (each tree has as vertices the homothety classes of lattices in the corresponding 2-dimensional local vector space). Emery states that no counterpart of Gelander's volume-linear cell bound is known for such groups; he circumvented this only for groups of type $A_1$ (rank-one factors), using the action on trees to write the group as an amalgamated product and induct, at the cost of constants depending on the norm of $S$.

The arithmetic motivation is algebraic $K$-theory: the Quillen $K$-groups $K_m(\mathcal{O}_F)$ are finitely generated with torsion-free rank given by Borel's regulator computation, and their torsion is a basic structural invariant; $K_2(\mathcal{O}_F)$ is finite and is called the tame kernel. Soulé (Doc. Math. 2003) proved general upper bounds for $|K_m(\mathcal{O}_F)_{\mathrm{tors}}|$ of size roughly exponential in $|D_F|^{1/2}$ and conjectured the optimal growth: there is a constant $C(m,d)$ with $\log |K_m(\mathcal{O}_F)_{\mathrm{tors}}| \le C(m,d) \log |D_F|$ for all number fields $F$ of degree $d$, of arbitrary signature — the direct analogue for higher $K$-groups of the logarithmic dependence on the discriminant in the class number bound. Calegari and Venkatesh (arXiv:1212.3847) constructed a surjection from the second homology of an adelic quotient attached to $\mathrm{PGL}_2$ over $F$, with coefficients in $\mathbb{Z}[1/6w_F]$ where $w_F$ is the number of roots of unity in $F$, onto $K_2(\mathcal{O}_F(S))$ tensored with the same ring; Emery combined this with his torsion-homology bounds to prove $\log |K_2(\mathcal{O}_F) \otimes \mathbb{Z}[1/6w_F]| \le C(d) |D_F|^2 (\log |D_F|)^{d-1}$ for totally imaginary $F$ of even degree $d$, and the analogous bound with a constant $C(d,N)$ for $S$-integer rings of $S$-norm at most $N$. These were the first bounds of this shape for $K_2$, but the paper itself records that the constants are not effective, that the exponents may be far from sharp, and that Soulé's conjectured logarithmic bound is out of reach of its methods.

Problem Statement

The common objective is uniform upper control of torsion growth — in the homology of arithmetic and $S$-arithmetic groups in terms of volume, and in the $K$-theory of rings of integers in terms of the discriminant — with Soulé's conjectural logarithmic bound as the optimal target. The source-faithful open components, each recorded as unresolved by Emery (Math. Z. 2014), are: (a) Soulé's conjecture: for each $m \ge 2$ and $d \ge 1$, does there exist a constant $C(m,d) < \infty$ such that

$$\log |K_m(\mathcal{O}_F)_{\mathrm{tors}}| \le C(m,d)\, \log |D_F|$$

for every number field $F$ of degree $d$ and arbitrary signature, where $D_F$ is the discriminant of $F$? (b) Removing the vanishing hypothesis from the torsion-homology bound: for a connected semisimple real algebraic group $G$ whose real points have trivial center and no compact factors, and for each $j \ge 1$, is there a constant $C_G$ with $\log |H_j(\Gamma)_{\mathrm{tors}}| \le C_G \, \mathrm{vol}(\Gamma \backslash X)$ for every irreducible nonuniform arithmetic lattice $\Gamma \subset G(\mathbb{R})$, without assuming $H_q(\Gamma, \mathbb{Q}) = 0$ for $q = 1, \ldots, j$? (c) The $S$-arithmetic geometric input: establish a Gelander-type structural bound — a cell complex of size linearly bounded by the volume — for $S$-arithmetic groups, i.e. for the quotients of the product of the symmetric space at the archimedean places with the Bruhat-Tits buildings at the finite places of $S$ by $S$-arithmetic subgroups, a counterpart that the paper records as unknown in general ("As far as we know, there is no known counterpart to Gelander's result for S-arithmetic groups"); the paper obtains bounds only for groups of type $A_1$, e.g. $\mathrm{SL}_2(\mathcal{O}_F(S))$ acting on $X_\infty \times \prod_{\mathfrak{p} \in S_f} X_\mathfrak{p}$ (the product of the symmetric space of $\mathrm{SL}_2(F \otimes_{\mathbb{Q}} \mathbb{R})$ with the Bruhat-Tits trees at the finite places in $S$, where $S_f$ denotes the set of finite (nonarchimedean) places in $S$ and $X_\mathfrak{p}$ is the Bruhat-Tits tree attached to $\mathfrak{p}$), using the action on trees to write the group as an amalgamated product and induct, at the cost of constants $C(d,N)$ depending on the norm $N_{F/\mathbb{Q}}(S)$. Component (c) supplies the missing geometric input that the Calegari-Venkatesh surjection from second homology onto $K_2$ converts into $K$-theoretic bounds — in the paper the rational-vanishing hypothesis of Theorem 1.1 is already verified for the groups relevant to its $K_2$ applications, so component (b) instead removes a standing hypothesis from the volume-linear torsion-homology bound itself — and the components jointly support the single objective of bounding torsion growth. Each of (a), (b) and (c) is an independently refutable open question: a resolution of the task is a proof or disproof of any one of them, and settling all three settles the objective in full but is not required.

References

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