diff --git a/Project.toml b/Project.toml
index 9a7d77b..4b3a75e 100644
--- a/Project.toml
+++ b/Project.toml
@@ -6,6 +6,7 @@ version = "0.4.2"
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+ThreadsX = "ac1d9e8a-700a-412c-b207-f0111f4b6c0d"
 
 [compat]
 AbstractAlgebra = "^0.9.0"
diff --git a/src/AutGroup.jl b/src/AutGroup.jl
index 46198cf..1ce4b00 100644
--- a/src/AutGroup.jl
+++ b/src/AutGroup.jl
@@ -258,7 +258,8 @@ function (==)(g::Automorphism{N}, h::Automorphism{N}) where N
     # cheap
     # if hashes differ, images must have differed as well
     hash(g) != hash(h) && return false
-    # equal elements, or possibly a hash conflict
+
+    # hashes equal, hence either equal elements, or a hash conflict
     begin
         if !img_computed
             img_task = Threads.@spawn img = compute_images(g)
@@ -271,6 +272,8 @@ function (==)(g::Automorphism{N}, h::Automorphism{N}) where N
         !img_computed && fetch(img_task)
         !imh_computed && fetch(imh_task)
     end
+
+    img != imh && @warn "hash collision in == :" g h
     return img == imh
 end
 
diff --git a/src/Groups.jl b/src/Groups.jl
index 3b79d37..dc4f5e6 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -64,24 +64,59 @@ Return vector of generators of `G`, as its elements.
 AbstractAlgebra.gens(G::AbstractFPGroup) = G.(G.gens)
 
 """
-    metric_ball(S::AbstractVector{<:GroupElem}
+    wlmetric_ball(S::AbstractVector{<:GroupElem}
         [, center=one(first(S)); radius=2, op=*])
 Compute metric ball as a list of elements of non-decreasing length, given the
 word-length metric on the group generated by `S`. The ball is centered at `center`
 (by default: the identity element). `radius` and `op` keywords specify the
 radius and multiplication operation to be used.
 """
-function generate_balls(S::AbstractVector{T}, center::T=one(first(S));
-        radius=2, op=*) where T<:Union{GroupElem, NCRingElem}
-    sizes = Int[]
-    B = [one(first(S))]
-    for i in 1:radius
-        BB = [op(i,j) for (i,j) in Base.product(B,S)]
-        B = unique([B; vec(BB)])
-        push!(sizes, length(B))
+
+function wlmetric_ball_serial(S::AbstractVector{T}; radius=2, op=*) where T<:Union{GroupElem, NCRingElem}
+    old = unique!([one(first(S)), S...])
+    sizes = [1, length(old)]
+    for i in 2:radius
+        new = collect(op(i,j) for (i,j) in Base.product(@view(old[sizes[end-1]:end]), S))
+        append!(old, new)
+        resize!(new, 0)
+        old = unique!(old)
+        push!(sizes, length(old))
     end
-    isone(center) && return B, sizes
-    return c.*B, sizes
+    return old, sizes[2:end]
+end
+
+function wlmetric_ball_thr(S::AbstractVector{T}; radius=2, op=*) where T<:Union{GroupElem, NCRingElem}
+    old = unique!([one(first(S)), S...])
+    sizes = [1, length(old)]
+    for r in 2:radius
+        begin
+            new = ThreadsX.collect(op(o, s) for o in @view(old[sizes[end-1]:end]) for s in S)
+            ThreadsX.foreach(hash, new)
+        end
+        append!(old, new)
+        resize!(new, 0)
+        old = ThreadsX.unique(old)
+        push!(sizes, length(old))
+    end
+    return old, sizes[2:end]
+end
+
+function wlmetric_ball_serial(S::AbstractVector{T}, center::T; radius=2, op=*) where T<:Union{GroupElem, NCRingElem}
+    E, sizes = metric_ball_serial(S, radius=radius, op=op)
+    isone(center) && return E, sizes
+    return c.*E, sizes
+end
+
+function wlmetric_ball_thr(S::AbstractVector{T}, center::T, radius=2, op=*) where T<:Union{GroupElem, NCRingElem}
+    E, sizes = metric_ball_thr(S, radius=radius, op=op)
+    isone(center) && return E, sizes
+    return c.*E, sizes
+end
+
+function wlmetric_ball(S::AbstractVector{T}, center::T=one(first(S);
+    radius=2, op=*, threading=true)
+    threading && return wlmetric_ball_thr(S, center, radius=radius, op=op)
+    return return wlmetric_ball_serial(S, center, radius=radius, op=op)
 end
 
 """
diff --git a/test/runtests.jl b/test/runtests.jl
index 25241ff..7e29894 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -13,7 +13,7 @@ using LinearAlgebra
       s = one(M); s[1,3] = 2; s[3,2] = -1;
 
       S = [w,r,s]; S = unique([S; inv.(S)]);
-      _, sizes = Groups.generate_balls(S, radius=4);
+      _, sizes = Groups.wlmetric_ball(S, radius=4);
       @test sizes == [7, 33, 141, 561]
    end