# Sofic one-relator groups from one-sided and single-block conjugators Date: 2026-09-07. All conjugations use `x^y=y^(-1)xy`. These are written proofs using named standard permanence theorems. They are not Lean certificates. The universal one-relator soficity question remains unresolved here. ## 1. Results and sources For a word `w in F(a,b)` and nonzero integers `l,k`, set G(w;l,k) = . **Theorem A (one-sided support).** Write `a_j=b^(-j)ab^j`. If `w=b^d V`, where `d>0` and `V` is a finite word in the `a_j` with `j>0`, then `G(w;l,k)` is sofic. **Theorem B (one a-block).** For every `p,q,s in Z` and every nonzero `l,k`, the group `G(b^p a^s b^q;l,k)` is sofic. **Corollary.** `G(b^(-1)ab^2;1,2)` is sofic. The last presentation is asked about in the introduction of [Berlai, A new family of sofic one-relator groups, arXiv:2502.05064v1](https://arxiv.org/html/2502.05064v1). That source proves two other families. The present arguments are deductions recorded in this repository, not assertions that the source contains these proofs. No claim of priority over all literature is made. We use amenable-edge amalgam permanence from Theorem 1 of [Elek--Szabo](https://arxiv.org/html/1010.3424v2), HNN permanence from Proposition 3.2 of [Ciobanu--Holt--Rees](https://arxiv.org/html/1212.2739v2), and the standard sofic-kernel/amenable-quotient permanence recalled and applied there. The companion [triangular-chain proof](triangular-bs-chain-proof-2026-09-07.md) checks the Baumslag-Solitar blocks and proves directed-colimit permanence directly from finite multiplication tables. Those inputs cover every permanence step below. ## 2. An exact kernel presentation Suppose `w=b^d V(a_j)` for some integer `d` and a finite indexed word `V`. Then, already in the free group, a^w = V^(-1) a_d V. Let `V_i` add `i` to every index in `V`, and define K = . The map shifting each index by one preserves all defining relations and has the inverse shift. Hence it induces an automorphism of `K`. Form the split extension `E` of `K` by an infinite cyclic element `b` satisfying b^(-1) a_i b = a_(i+1). This is a semidirect product: in the usual convention `b h b^(-1)` is the action of the generator, that action is the inverse shift. Its kernel under `b->1,a_i->0` is exactly `K`. In this presentation of `E`, eliminate every `a_i` other than `a_0` using `a_i=b^(-i)a_0b^i`. All kernel relations become conjugates of the index-zero relation. The remaining presentation is exactly `G(w;l,k)` after naming `a_0` as `a`. Conversely these substitutions define a homomorphism from the displayed presentation of `E` to `G(w;l,k)`, and the assignment `a->a_0,b->b` gives its inverse. This verifies the presentation equality and kernel identification in both directions. Thus `G(w;l,k)` has sofic kernel if the displayed `K` is sofic, and its quotient is `Z`. Proving `K` sofic will therefore suffice. ## 3. Proof of Theorem A and the explicit corollary Under Theorem A's hypotheses, `d>0` and every index of `V_i` exceeds `i`. The conjugator `V_i^(-1) a_(i+d) V_i` meets the triangular-chain theorem with `j_i=i+d` and `s_i=1`. Hence `K` is sofic. The extension of Section 2 has amenable quotient `Z`, so `G(w;l,k)` is sofic. This proves Theorem A. For the requested mixed-conjugator example, direct free reduction gives b^(-1)ab^2 = b a_2, a^(b^(-1)ab^2) = a_2^(-1) a_1 a_2. Take `d=1,V=a_2`. Explicitly, the kernel is . For any finite set of relation indices, process them in decreasing order. The only new generator at step `i` is `a_i`; the conjugator uses later generators and has infinite order because it is conjugate to `a_(i+1)`. Thus this instance uses precisely the cyclic-amalgam induction already proved. Taking `l=1,k=2` proves the corollary. ## 4. Three ways to attach one generator We need a slightly more flexible construction for Theorem B. Let `H` be a sofic group containing the old elements described below, each of infinite order. Assume `l,k,s` are nonzero. Consider the single relation (z^(-s) y z^s)^(-1) x^l (z^(-s) y z^s) = x^k. (R) Exactly one of `x,y,z` will be a new generator. There is no assertion that the two old elements generate a free subgroup. ### 4.1. New x The old element `c=z^(-s) y z^s` has infinite order. Form `H *_(c=t) BS(l,k)`, using `BS(l,k)=`. Both edge maps are injective. Both factors embed, the amalgam is sofic, and the new `x` has infinite order. Eliminating `t=c` gives exactly `(R)`. ### 4.2. New y Set `v=z^s x z^(-s)` in `H`, an infinite-order conjugate of `x`. Conjugating `(R)` by `z^s` gives the equivalent relation y^(-1) v^l y = v^k. Adjoin `y` as an HNN stable letter identifying `` with `` by `v^(ln)->v^(kn)`. The nonzero exponents make both maps injective. The associated subgroups are cyclic, hence amenable, so the extension is sofic and embeds `H`. Its stable letter `y` has infinite order, as shown by the homomorphism sending `H` to zero and `y` to one in `Z`. Conjugating back gives exactly `(R)`. ### 4.3. New z First form J = H *_(x=X) . The element `x` in `H` and the base generator `X` of the Baumslag-Solitar block have infinite order. Therefore the edge maps are injective; `J` is sofic and embeds both factors. In particular `y` and `t` have infinite order in `J`. Next adjoin a stable letter `h` with `h^(-1)y h=t`. This is an HNN extension of `J` along the infinite cyclic subgroups `` and ``. It is sofic, embeds `J`, and its stable letter `h` has infinite order. Finally form the amalgam with an infinite cyclic group ``, identifying `h` with `z^s`. Both edge maps are injective, including when `s<0`. The final group is sofic and embeds the preceding group and ``. Thus the old generators and the new `z` all retain infinite order. After eliminating `X`, the new relations are exactly t^(-1)x^l t=x^k, h^(-1)y h=t, h=z^s. Eliminating `h` and `t` produces exactly `(R)`. This last elimination is essential: the construction proves equality with the desired presented group, not merely a homomorphism to a sofic overgroup. ## 5. Distinct indices in Theorem B Put `d=p+q`. Free reduction gives w=b^p a^s b^q = b^d a_q^s, a^w = a_q^(-s) a_d a_q^s. The kernel of Section 2 therefore has one relation `(R_i)` of form `(R)` for each integer `i`, with x=a_i, y=a_(i+d), z=a_(i+q). Assume first that `0,d,q` are pairwise distinct and `s!=0`. Set `m=min(0,d,q)`. Every relator has a unique least-index generator `a_(i+m)`. The other two generator indices are strictly larger. Fix a finite set `D` of relation indices. Begin with the free group on all generator names except `{a_(i+m): i in D}`. Process `D` in decreasing order of `i`. At step `i` the least-index generator is new; both other names have already been adjoined or belonged to the initial free group. Maintain the invariant that the current group is sofic, has exactly the relations processed so far, and every named generator has infinite order. If `m=0`, apply Section 4.1; if `m=d`, apply Section 4.2; if `m=q`, apply Section 4.3. Each construction preserves that invariant. This proves that every finite-relator truncation on the full generator set is sofic. Their directed colimit is the kernel `K`. The finite-table limit proof in the companion artifact applies without requiring the transition maps to be injective. Hence `K` is sofic, and extension by `Z` proves Theorem B in the distinct-index case. ## 6. All degenerate cases ### 6.1. Pure b-powers If `w=b^d` with `d!=0`, the kernel relators are a_(i+d)^(-1) a_i^l a_(i+d)=a_i^k. For `d>0` the triangular theorem applies. For `d<0`, replace `b` by `b^(-1)` and apply the positive case. This proves `G(b^d;l,k)` sofic. If `w` is any power of `a`, including the empty word, then `a^w=a` and the relation reduces to `a^(l-k)=1`. The group is the free product of `Z` with a cyclic group (infinite cyclic if `l=k`), so is sofic. ### 6.2. Vanishing block or end exponent If `s=0`, use Section 6.1 with `d=p+q`. If `p=0`, then `w=a^s b^q` and `a^w=a_q`, so use the same section. If `q=0`, then `a^w=a^(-s)a_p a^s`. Conjugating the defining relator by `a^s`, which commutes with both `a^l` and `a^k`, reduces it to the pure-power case `w=b^p`. ### 6.3. d=0 with q and s nonzero The only remaining index collision is `p+q=0`, with `q,s!=0`. We first construct the block D_s = . Start from `BS(l,k)=`. Adjoin an HNN letter `h` with `h^(-1)x h=t`, and then adjoin an `s`-th root `z` of `h` by a cyclic amalgam. All the edge maps are injective because `x,t,h` have infinite order and `s!=0`. Soficity follows at every step. Eliminating `h,t` gives exactly `D_s`, and both `x,z` have infinite order. The kernel in this collision case has one `D_s` relation on each pair `(a_i,a_(i+q))`. In a finite truncation, process the least-index generators in decreasing order. Attach a copy of `D_s` along the old generator: along its `z` if `q>0`, or along its `x` if `q<0`. The shared subgroup is infinite cyclic, both factors embed, and the new generator has infinite order. This again proves the finite truncations sofic. The limit and cyclic extension arguments finish this case. The cases in Sections 5 and 6 exhaust all integers `p,q,s`. This completes Theorem B. QED. ## 7. What this changes, and what remains The construction proves a positive family beyond the rank-one Magnus overlap case. For `w=b^(-1)ab^2`, the ordinary three-generator window has a rank-two free overlap, but its alternative construction above uses only cyclic edges. Rank at least two is therefore an obstruction to the generic permanence shortcut, not a reason to leave every such example open. For a general word `w` containing several `a`-blocks, the least indexed letter may occur in several different places of the conjugating word. It then need not fit any of the three single-new-generator constructions in Section 4. For a general one-relator presentation, even the nested Baumslag shape need not occur. No argument here supplies either missing reduction. The claims `one-relator-groups-sofic`, `magnus-staggered-chain-sofic`, and `linton-shell-sofic` remain open. ## 8. Reproducible checks Run from the repository root: python3 experiments/one_relator_cyclic_peeling_check.py --json bin/cairn check --changed The deterministic [replay report](one-relator-cyclic-peeling-replay-2026-09-07.json) records 3,920 single-block parameter choices, 11,760 shifted Schreier identities, 1,800 one-sided word identities, 64 sets of attachment and auxiliary-generator elimination identities, and 360 finite dependency schedules. All seven cases of Theorem B are exercised. Five negative controls reject nonkernel words and unsupported one-sided hypotheses. The word checks independently compare a left-to-right exponent-height scan against the displayed kernel formulas, and decode the scanned words back to the original free-group words. They establish exact finite identities and scheduling conditions. Infinite-order preservation, the exact kernel identification, the limit argument, and sofic permanence are justified by the written mathematical proof, not by these computations. Cairn verifies the research graph's consistency; it is not a proof assistant.