Profinitely Solitary Lattices

• Profinitely solitary groups are finitely generated, residually finite groups whose profinite completion uniquely determines their abstract commensurability class.
• The concept leverages arithmetic lattice properties, Margulis arithmeticity, and superrigidity to connect finite local data with the global structure of the group.
• Adelic superrigidity and Galois cohomology are key, revealing obstructions and rigidity phenomena that underpin profinite solitude.

“Profinitely solitary” designates a rigidity property of a finitely generated, residually finite group, or more specifically of an arithmetic lattice, asserting that the commensurability class of its profinite completion determines its ordinary abstract commensurability class. In the higher-rank setting, the notion is tied to Margulis arithmeticity and superrigidity, the congruence subgroup property (CSP), and the structure of finite adeles. The modern theory shows that profinite solitude is governed not merely by the ambient real or complex Lie group, but by the adelic form of the underlying algebraic group and by Galois-cohomological obstructions to changing the real form while preserving all finite local data (Kammeyer et al., 2020, Kammeyer et al., 2023). A stronger notion, absolute profinite solitude, asks for the same determination among all finitely generated residually finite groups rather than only among lattices in a fixed Lie group; in higher rank, this stronger form often depends on a Grothendieck rigidity conjecture and fails systematically in many cocompact cases (Kammeyer, 2022).

1. Definitions and basic distinctions

Let $\Gamma$ be a finitely generated group. Its profinite completion is

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$

If $\Gamma$ is residually finite, the natural map $\Gamma\to\widehat{\Gamma}$ is injective (Kammeyer, 2022, Kammeyer et al., 2023).

Two groups $A,B$ are commensurable, written $A\sim B$, if they contain isomorphic finite-index subgroups (Kammeyer, 2022, Kammeyer et al., 2023). Two groups $\Gamma_1,\Gamma_2$ are profinitely commensurable if $\widehat{\Gamma}_1$ and $\widehat{\Gamma}_2$ contain isomorphic open subgroups (Kammeyer, 2022). In the lattice setting, one also says that two lattices $\Gamma,\Delta\le G(K)$ are profinitely commensurable if an open subgroup of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$0 is isomorphic to an open subgroup of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$1 (Kammeyer et al., 2020).

A finitely generated, residually finite group $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$2 is profinitely solitary if every finitely generated, residually finite group $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$3 with

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$4

satisfies $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$5 (Kammeyer et al., 2023). For a lattice $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$6, a weaker ambient version is standard: $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$7 is profinitely solitary in $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$8 if every other lattice $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$9 with $\Gamma$0 satisfies $\Gamma$1 (Kammeyer, 2022). In the formulation of higher-rank lattices over $\Gamma$2, a lattice is profinitely solitary precisely when its profinite commensurability class in $\Gamma$3 coincides with its ordinary commensurability class (Kammeyer et al., 2020).

The absolute form is stronger. A lattice $\Gamma$4 is absolutely profinitely solitary if for every finitely generated residually finite group $\Gamma$5,

$\Gamma$6

This removes the restriction that comparison groups be lattices in the same Lie group (Kammeyer, 2022).

The main conceptual distinction is therefore between three levels of rigidity: equality of profinite completions, profinite commensurability, and abstract commensurability. The theory of profinitely solitary lattices studies when the first or second level forces the third.

2. Higher-rank arithmeticity and the adelic viewpoint

The subject is formulated for lattices in connected simple Lie groups, especially in higher rank. In the setup of adelic superrigidity, one takes $\Gamma$7 to be either $\Gamma$8 or $\Gamma$9, and $\Gamma\to\widehat{\Gamma}$0 a connected, simply connected, absolutely almost simple linear algebraic $\Gamma\to\widehat{\Gamma}$1-group of $\Gamma\to\widehat{\Gamma}$2-rank $\Gamma\to\widehat{\Gamma}$3, or more generally a group for which Margulis-type superrigidity holds (Kammeyer et al., 2020). A lattice $\Gamma\to\widehat{\Gamma}$4 is a discrete subgroup of finite covolume (Kammeyer et al., 2020).

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\Gamma\to\widehat{\Gamma}$5 or $\Gamma\to\widehat{\Gamma}$6 (Kammeyer et al., 2020). In the later absolute-solitude work, the standing assumption is that $\Gamma\to\widehat{\Gamma}$7 is connected and simple and that all lattices are arithmetic by Margulis’s theorem (Kammeyer, 2022).

The finite-adelic perspective is essential. For a number field $\Gamma\to\widehat{\Gamma}$8, with $\Gamma\to\widehat{\Gamma}$9 the set of finite places, $A,B$0 the completion at $A,B$1, and

$A,B$2

the ring of finite adeles, arithmetic subgroups of a simply-connected, absolutely almost simple $A,B$3-group are controlled by open compact subgroups of $A,B$4 through strong approximation and CSP in higher rank (Kammeyer et al., 2020, Kammeyer et al., 2023). This is why profinite completion, which records all finite quotients simultaneously, interacts naturally with adele rings and adelic points.

A central consequence is that profinite information is not merely “local modulo all finite quotients” in an abstract sense; in higher rank it is tied to the finite adelic form of the ambient algebraic group. This suggests that the natural equivalence relation behind profinite rigidity is adelic isomorphism rather than only real or complex isomorphism.

3. Adelic superrigidity and profinite commensurability

The principal structural result is an adelic refinement of Margulis superrigidity. Let $A,B$5 be a number field, let $A,B$6 be a connected, simply connected, absolutely almost simple $A,B$7-group which is algebraically superrigid in the sense that every homomorphism from an arithmetic subgroup into the $A,B$8-points of another absolutely almost simple $A,B$9-group extends, up to a central factor, to a homomorphism of algebraic groups (Kammeyer et al., 2020).

The adelic superrigidity theorem states that if $A\sim B$0 is an arithmetic subgroup, $A\sim B$1 is a second number field, $A\sim B$2 is a connected absolutely almost simple $A\sim B$3-group, and

$A\sim B$4

has image with non-empty interior in the locally compact group $A\sim B$5, then there exist:

1. an injective map of places $A\sim B$6,
2. for each $A\sim B$7, an isomorphism of local fields $A\sim B$8, inducing an injective ring homomorphism $A\sim B$9,
3. a homomorphism of group schemes over $\Gamma_1,\Gamma_2$0,

$\Gamma_1,\Gamma_2$1

1. and a finite-order factor $\Gamma_1,\Gamma_2$2

such that

$\Gamma_1,\Gamma_2$3

for every $\Gamma_1,\Gamma_2$4 (Kammeyer et al., 2020).

The data are uniquely determined, and one always has $\Gamma_1,\Gamma_2$5, with equality precisely when $\Gamma_1,\Gamma_2$6 is a bijection and $\Gamma_1,\Gamma_2$7 is an isomorphism $\Gamma_1,\Gamma_2$8 (Kammeyer et al., 2020). In particular, any profinite-level isomorphism between two arithmetic lattices forces an adelic isomorphism between their parent algebraic groups (Kammeyer et al., 2020).

The proof proceeds in three stages. First, at each finite place one applies classical Margulis superrigidity to obtain local homomorphisms $\Gamma_1,\Gamma_2$9 over uniquely determined field isomorphisms $\widehat{\Gamma}_1$0, while a Baire-category argument fixes the place matching $\widehat{\Gamma}_1$1 by forcing local degree equalities (Kammeyer et al., 2020). Second, one patches the local data to the adele ring using an algebraic-geometry gluing lemma, producing a global morphism of group schemes over the adeles (Kammeyer et al., 2020). Third, one tracks the finite-image central factor and derives uniqueness of all data from uniqueness at the local stage (Kammeyer et al., 2020).

This theorem yields the profinite commensurability classification for arithmetic lattices. If $\widehat{\Gamma}_1$2 and $\widehat{\Gamma}_1$3 are absolutely almost simple algebraically superrigid groups over number fields $\widehat{\Gamma}_1$4 and $\widehat{\Gamma}_1$5, and $\widehat{\Gamma}_1$6 and $\widehat{\Gamma}_1$7 are arithmetic subgroups with $\widehat{\Gamma}_1$8, then there is a ring isomorphism $\widehat{\Gamma}_1$9, an isomorphism of group schemes over $\widehat{\Gamma}_2$0, and a finite-image central factor compatible with the congruence completions (Kammeyer et al., 2020). Conversely, an adelic isomorphism of the underlying algebraic groups is equivalent to profinite commensurability of their lattices (Kammeyer et al., 2020).

When $\widehat{\Gamma}_2$1 has CSP, the resulting classification is:

$\widehat{\Gamma}_2$2

Here a $\widehat{\Gamma}_2$3-arithmetic pair is a pair $\widehat{\Gamma}_2$4 with $\widehat{\Gamma}_2$5 or $\widehat{\Gamma}_2$6 a number subfield, $\widehat{\Gamma}_2$7 a simply connected anisotropic-outside-$\widehat{\Gamma}_2$8 $\widehat{\Gamma}_2$9-group $\Gamma,\Delta\le G(K)$0-isogenous to $\Gamma,\Delta\le G(K)$1, and “locally isomorphic” means isomorphic over $\Gamma,\Delta\le G(K)$2 (Kammeyer et al., 2020).

A common misconception is that an isomorphism of profinite completions should directly force isomorphism, or at least commensurability, of the original higher-rank lattices. The classification shows that the immediate invariant is adelic: ordinary commensurability follows only when the relevant adelic isomorphism class contains a single commensurability class.

4. Galois cohomology and the finite splitting principle

For split arithmetic Chevalley groups, profinite solitude is analyzed through Galois cohomology. Let $\Gamma,\Delta\le G(K)$3 be a number field, $\Gamma,\Delta\le G(K)$4 its ring of integers, and $\Gamma,\Delta\le G(K)$5 a simply-connected, absolutely almost simple, $\Gamma,\Delta\le G(K)$6-split linear algebraic group (Kammeyer et al., 2023). An arithmetic subgroup is a subgroup commensurable with $\Gamma,\Delta\le G(K)$7, and by strong approximation and CSP in higher rank, $\Gamma,\Delta\le G(K)$8 is commensurable with an open compact subgroup of $\Gamma,\Delta\le G(K)$9 (Kammeyer et al., 2023).

The obstruction to profinite solitude comes from non-split inner twists of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$00 that are trivial at all finite places of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$01 (Kammeyer et al., 2023). These twists are encoded by the exact sequences

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$02

and

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$03

where $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$04 is the Dynkin diagram (Kammeyer et al., 2023).

The key map is the adelic-restriction map

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$05

defined by restriction to all non-archimedean completions (Kammeyer et al., 2023). Its kernel classifies $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$06-forms of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$07 which split at every finite place of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$08. Such a form gives rise to an arithmetic subgroup whose profinite completion is isomorphic to that of the split group, and hence $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$09 is profinitely solitary precisely when $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$10 (Kammeyer et al., 2023).

The cohomological analysis reduces this to a question about the center. By a diagram chase,

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$11

and under mild hypotheses one has $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$12; moreover the connecting map $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$13 is trivial by Hilbert’s Theorem 90 (Kammeyer et al., 2023). Thus $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$14 provided the obstruction map

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$15

is injective (Kammeyer et al., 2023).

When $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$16, the Albert–Brauer–Hasse–Noether exact sequence implies that restriction to all finite places is injective as soon as either $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$17 has at most one real place or $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$18 is odd, which covers types $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$19 (Kammeyer et al., 2023).

This yields the finite splitting principle: for a simply-connected, absolutely almost simple, split $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$20-group $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$21, the adelic restriction map is injective if and only if one of the following holds (Kammeyer et al., 2023):

• $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$22 is totally imaginary;
• $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$23 has exactly one real place and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$24 has type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$25 or $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$26;
• $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$27 is arbitrary and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$28 has type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$29.

In all other cases one exhibits a nontrivial inner twist of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$30 which is split at all finite places but differs at $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$31 (Kammeyer et al., 2023). This is the basic mechanism by which non-solitude arises: the finite adelic data remain unchanged while the archimedean form changes.

Under the additional assumptions that $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$32 is locally determined, that $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$33 has CSP, and that $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$34 has no proper Grothendieck subgroup, Kammeyer–Spitler obtain a positive theorem for arithmetic subgroups of split groups in a range of cases. These include: totally imaginary $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$35 with the stated exclusion for imaginary quadratic type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$36; one real place and type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$37 or $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$38 in the stated ranges; arbitrary $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$39 and type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$40; $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$41 and types $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$42; and type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$43 over the signatures $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$44 (Kammeyer et al., 2023). If these conditions fail, and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$45 is locally determined and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$46 is not of type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$47 and not of type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$48 when $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$49, then no arithmetic subgroup of $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$50 is profinitely solitary (Kammeyer et al., 2023).

5. Absolute profinite solitude in higher-rank Lie groups

Absolute profinite solitude strengthens the lattice-in-$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$51 notion by comparing with all finitely generated residually finite groups (Kammeyer, 2022). The central input is a conjectural Grothendieck rigidity statement:

Conjecture A. If $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$52 is a connected, almost simple algebraic group over a number field $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$53 with CSP, then every arithmetic subgroup $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$54 is Grothendieck rigid: every injective homomorphism

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$55

that induces an isomorphism $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$56 is surjective (Kammeyer, 2022).

A second hypothesis appears for one exceptional family:

Conjecture B. All lattices in the real form $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$57 have the congruence subgroup property (Kammeyer, 2022).

The mechanism is as follows. Spitler’s theorem implies that if a finitely generated residually finite group $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$58 satisfies $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$59, then $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$60 embeds as a subgroup of some arithmetic group $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$61 which is adelically isomorphic to the adele points of the form defining $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$62 (Kammeyer, 2022). CSP for both forms upgrades this to an injective map on profinite completions, and Conjecture A is then used to conclude that an injective map of arithmetic groups inducing an isomorphism on profinite completions is already an isomorphism on the discrete level (Kammeyer, 2022). This yields absolute solitude in the noncocompact cases identified in the theorem.

The main results divide sharply by Lie type and by cocompact versus noncocompact lattices (Kammeyer, 2022):

• In simple complex Lie groups of Cartan type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$63, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$64, or $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$65, cocompact lattices are not absolutely profinitely solitary, whereas noncocompact lattices are absolutely profinitely solitary assuming Conjecture A.
• In the real groups $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$66, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$67, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$68, and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$69, one has:
  • in $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$70, cocompact implies not absolutely solitary, while noncocompact plus Conjecture A implies absolutely solitary;
  • in $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$71, cocompact implies not absolutely solitary, while noncocompact plus Conjecture B implies not absolutely solitary;
  • in $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$72 and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$73, lattices are never absolutely profinitely solitary.
• In higher-rank real Lie groups whose complexification has Cartan type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$74, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$75, or $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$76, under mild bounds on $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$77, lattices are never absolutely profinitely solitary; moreover, in every such type there are both cocompact and noncocompact lattices with the same profinite completion yet not commensurable.
• For $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$78 with $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$79, every noncocompact lattice is absolutely profinitely solitary if Conjecture A holds.

The contrast between cocompact and noncocompact lattices is a recurring feature. In the exceptional complex groups, cocompact arithmetic lattices fail absolute solitude because one can construct the non-isomorphic $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$80-split form over the same number field and thereby obtain a non-commensurable lattice with the same profinite completion (Kammeyer, 2022). In the noncocompact split case, the field is imaginary quadratic and arithmetically solitary, forcing any other candidate form to live over the same field; CSP plus Grothendieck rigidity then shows commensurability (Kammeyer, 2022).

For $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$81, Galois cohomology shows that there are no nontrivial $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$82-twists that are locally trivial at all primes and at $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$83, so any adelic partner is actually isomorphic over $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$84; this gives absolute solitude for noncocompact lattices under Conjecture A (Kammeyer, 2022). By contrast, every cocompact form $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$85 with $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$86 arises from a nontrivial $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$87-twist and gives a counterexample (Kammeyer, 2022).

A plausible implication is that absolute profinite solitude is considerably more sensitive than ordinary profinite solitude to the existence of hidden arithmetic embeddings and self-embeddings detectable only through Grothendieck rigidity.

6. Examples, consequences, and limits of the invariant

Several explicit classes of profinitely solitary lattices are identified in the adelic-superrigidity framework. For the simply connected real forms of type $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$88, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$89, $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$90, and $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$91, non-existence of outer automorphisms implies that every totally real subfield $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$92 gives rise to a unique $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$93-arithmetic pair $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$94 and hence to a unique commensurability class $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$95 (Kammeyer et al., 2020). All these adelic groups are isomorphic over $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$96, so all lattices $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$97 are profinitely commensurable; nevertheless,

$$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$98

(Kammeyer et al., 2020).

If $$\widehat{\Gamma}=\varprojlim_{N\triangleleft \Gamma,\;[\Gamma:N]<\infty}\Gamma/N.$$99 is not of type $\Gamma$00, $\Gamma$01, or $\Gamma$02 and $\Gamma$03, then outer automorphisms again vanish, and every $\Gamma$04-arithmetic lattice in $\Gamma$05 is profinitely solitary (Kammeyer et al., 2020). In particular, every non-cocompact lattice in such a $\Gamma$06 is profinitely solitary (Kammeyer et al., 2020).

At the opposite extreme, rank one without CSP behaves differently. In groups $\Gamma$07 or $\Gamma$08 without CSP, there are non-commensurable lattices with isomorphic profinite completions (Kammeyer et al., 2020). This marks a sharp boundary: the higher-rank theory relies fundamentally on arithmeticity, superrigidity, and CSP.

Additional structural consequences appear already at the level of profinite completions. For an arithmetic lattice $\Gamma$09, the kernel of the natural map

$\Gamma$10

is a characteristic subgroup of $\Gamma$11 (Kammeyer et al., 2020). Moreover, every dense subfield $\Gamma$12 or $\Gamma$13 of finite degree with the right anisotropy data occurs in a $\Gamma$14-arithmetic pair, so $\Gamma$15 always has infinitely many commensurability classes of lattices, yet for exceptional $\Gamma$16 only finitely many profinite ones (Kammeyer et al., 2020).

The split Chevalley-group work provides further concrete examples. Under Grothendieck rigidity, profinitely solitary examples include $\Gamma$17 for $\Gamma$18, $\Gamma$19 for $\Gamma$20, and $\Gamma$21, together with several examples over nontrivial number rings listed explicitly in the paper (Kammeyer et al., 2023). Non-solitary examples include $\Gamma$22, $\Gamma$23, $\Gamma$24, $\Gamma$25, and $\Gamma$26, again with further examples in types $\Gamma$27, $\Gamma$28, $\Gamma$29, and $\Gamma$30 (Kammeyer et al., 2023).

These examples show that profinite completion is neither uniformly weak nor uniformly complete as an invariant. Its effectiveness depends on whether finite local data determine the global form and whether the ambient arithmetic group satisfies the rigidity inputs needed to descend from the profinite to the discrete level.

7. Conceptual significance and open problems

The theory identifies a precise relationship among superrigidity, adelic isomorphism, CSP, and Galois cohomology. Adelic superrigidity unifies and strengthens classical rigidity results of Margulis by showing that a profinite-level isomorphism forces an adelic isomorphism of the parent algebraic groups (Kammeyer et al., 2020). Galois cohomology then determines whether that adelic isomorphism class contains one commensurability class or several (Kammeyer et al., 2023). Absolute solitude adds a second layer: whether every finitely generated residually finite group with the same profinite completion must actually arise from the same commensurability class, a step currently linked to Grothendieck rigidity (Kammeyer, 2022).

Two open problems dominate the current picture. The first is Conjecture A, Grothendieck rigidity for higher-rank arithmetic groups with CSP (Kammeyer, 2022). The positive absolute-solitude results for noncocompact lattices in $\Gamma$31, $\Gamma$32, $\Gamma$33, $\Gamma$34, and $\Gamma$35 depend on this conjecture (Kammeyer, 2022). The second is Conjecture B on CSP for $\Gamma$36, which is used to settle the $\Gamma$37 case in the stated form (Kammeyer, 2022).

The present state of the subject therefore supports a nuanced conclusion. In higher rank, profinite data are often strong enough to recover the adelic form and sometimes the commensurability class itself; in certain split and noncocompact settings they are expected to determine even the absolute commensurability class among all finitely generated residually finite groups. But cocompact lattices, nontrivial inner twists trivial at all finite places, and the absence of CSP or Grothendieck rigidity provide systematic obstructions. This suggests that “profinitely solitary” is best understood not as a universal rigidity phenomenon, but as a precise arithmetic condition measuring when finite local behavior determines global commensurability.